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A concept formalizing a number of algebraic properties of collections of morphism between mathematical objects of the same type (sets, topological spaces, groups, etc.) under the condition that these collections contain the identity mappings and are closed with respect to successive composition (or product) of mappings. A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207401.png" /> consists of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207402.png" />, whose elements are called objects of the category, and a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207403.png" />, the elements of which are called morphisms of the category. These classes must satisfy the following conditions:
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1) to each ordered pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207404.png" /> is associated a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207405.png" /> (also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207407.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207408.png" />) of members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c0207409.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074011.png" /> is called the source, or domain (of definition), of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074013.png" /> the target, or range (of values), or codomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074014.png" />; instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074015.png" /> one often writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074017.png" />;
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2) each morphism of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074018.png" /> belongs to only one set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074019.png" />;
+
A concept formalizing a number of algebraic properties of collections of morphism between mathematical objects of the same type (sets, topological spaces, groups, etc.) under the condition that these collections contain the identity mappings and are closed with respect to successive composition (or product) of mappings. A category  $  \mathfrak C $
 +
consists of a class  $  \mathop{\rm Ob}  \mathfrak C $,
 +
whose elements are called objects of the category, and a class  $  \mathop{\rm Mor}  \mathfrak C $,
 +
the elements of which are called morphisms of the category. These classes must satisfy the following conditions:
  
3) in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074020.png" />, the following partial composition (product) rule is given: The product of two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074022.png" /> is defined if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074023.png" />, in which case it belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074024.png" />; the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074026.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074027.png" />, or, depending on one's choice of conventions, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074028.png" />;
+
1) to each ordered pair of objects  $  A , B $
 +
is associated a set  $  H _ {\mathfrak C }  ( A , B ) $(
 +
also denoted by  $  \mathfrak C ( A , B ) $,
 +
$  \mathop{\rm Hom} ( A , B ) $,  
 +
or  $  H ( A , B ) $)  
 +
of members of $  \mathop{\rm Mor} $.  
 +
If  $  \alpha \in H ( A , B ) $,
 +
then  $  A $
 +
is called the source, or domain (of definition), of the morphism  $  \alpha $,
 +
and $  B $
 +
the target, or range (of values), or codomain of  $  \alpha $;
 +
instead of $  \alpha \in H ( A , B ) $
 +
one often writes  $  \alpha : A \rightarrow B $
 +
or  $  A \rightarrow  ^  \alpha  B $;
  
4) for any morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074031.png" />, the associativity law holds:
+
2) each morphism of the category  $  \mathfrak C $
 +
belongs to only one set  $  H _ {\mathfrak C }  ( A , B ) $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074032.png" /></td> </tr></table>
+
3) in the class $  \mathop{\rm Mor}  \mathfrak C $,
 +
the following partial composition (product) rule is given: The product of two morphisms  $  \alpha :  A \rightarrow B $
 +
and  $  \beta : C \rightarrow D $
 +
is defined if and only if  $  B = C $,
 +
in which case it belongs to  $  H ( A , D ) $;  
 +
the product of  $  \alpha $
 +
and  $  \beta $
 +
is denoted by  $  \alpha \beta $,
 +
or, depending on one's choice of conventions, by  $  \beta \alpha $;
  
5) each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074033.png" /> contains a (distinguished) morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074036.png" /> for any morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074038.png" />; the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074039.png" /> are called units, identities or ones.
+
4) for any morphisms  $  \alpha :  A \rightarrow B $,
 +
$  \beta :  B \rightarrow C $
 +
and  $  \gamma :  C \rightarrow D $,
 +
the associativity law holds:
 +
 
 +
$$
 +
( \alpha \beta ) \gamma  =  \alpha ( \beta \gamma ) ;
 +
$$
 +
 
 +
5) each set $  H _ {\mathfrak C }  ( A , A ) $
 +
contains a (distinguished) morphism $  1 _ {A} $
 +
such that $  \alpha \cdot 1 _ {A} = \alpha $
 +
and $  1 _ {A} \cdot \beta = \beta $
 +
for any morphisms $  \alpha : X \rightarrow A $
 +
and $  \beta : A \rightarrow Y $;  
 +
the morphisms $  1 _ {A} $
 +
are called units, identities or ones.
  
 
The notion of a class occurring in the definition of a category presupposes the use of axioms from set theory which distinguish between sets and classes. The most commonly used is the axiom scheme of Gödel–Bernays–von Neumann.
 
The notion of a class occurring in the definition of a category presupposes the use of axioms from set theory which distinguish between sets and classes. The most commonly used is the axiom scheme of Gödel–Bernays–von Neumann.
  
Sometimes, in the definition of a category, it is not required that the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074040.png" /> be sets. Sometimes, instead of using classes, one assumes the existence of a universal set and requires that the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074042.png" /> belong to a fixed universal set.
+
Sometimes, in the definition of a category, it is not required that the classes $  H ( A , B ) $
 +
be sets. Sometimes, instead of using classes, one assumes the existence of a universal set and requires that the classes $  \mathop{\rm Ob}  \mathfrak C $
 +
and $  \mathop{\rm Mor}  \mathfrak C $
 +
belong to a fixed universal set.
  
Since there is a bijective correspondence between the identities of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074043.png" /> and the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074044.png" />, a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074045.png" /> can be defined as a class of morphisms with a partial product satisfying additional conditions (see, for example, [[#References|[6]]], [[#References|[9]]]).
+
Since there is a bijective correspondence between the identities of a category $  \mathfrak C $
 +
and the class $  \mathop{\rm Ob}  \mathfrak C $,  
 +
a category $  \mathfrak C $
 +
can be defined as a class of morphisms with a partial product satisfying additional conditions (see, for example, [[#References|[6]]], [[#References|[9]]]).
  
 
The notion of a category was introduced in 1945 [[#References|[8]]]. The origins of category theory and the initial stimulus for its development came from algebraic topology. Subsequent investigation revealed the unifying role of the notion of a category and the notion of a [[Functor|functor]] related to it for many branches of mathematics.
 
The notion of a category was introduced in 1945 [[#References|[8]]]. The origins of category theory and the initial stimulus for its development came from algebraic topology. Subsequent investigation revealed the unifying role of the notion of a category and the notion of a [[Functor|functor]] related to it for many branches of mathematics.
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===Examples of categories.===
 
===Examples of categories.===
  
 +
1) The category of sets  $  \mathop{\rm Ens} $.
 +
The class  $  \textrm{ Ob  Ens  } $
 +
consists of all sets; the class  $  \textrm{ Mor  Ens  } $
 +
consists of all functions between sets; composition of morphism is composition of functions (see [[Sets, category of|Sets, category of]]).
  
1) The category of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074046.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074047.png" /> consists of all sets; the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074048.png" /> consists of all functions between sets; composition of morphism is composition of functions (see [[Sets, category of|Sets, category of]]).
+
2) The category of topological spaces $  \mathop{\rm Top} $(
 
+
or $  \mathfrak T  $).  
2) The category of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074049.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074050.png" />). The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074051.png" /> consists of all topological spaces, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074052.png" /> of all continuous mappings, while composition is again composition of mappings.
+
The class $  \textrm{ Ob  Top  } $
 +
consists of all topological spaces, the class $  \textrm{ Mor  Top  } $
 +
of all continuous mappings, while composition is again composition of mappings.
  
3) The category of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074053.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074054.png" />). The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074055.png" /> consists of all groups, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074056.png" /> of all group homomorphisms, and composition again is composition of homomorphisms (see [[Category of groups|Category of groups]]). One defines in a similar fashion the categories of vector spaces over some field, the category of rings, etc.
+
3) The category of groups $  \mathop{\rm Gr} $(
 +
or $  \mathfrak G $).  
 +
The class $  \textrm{ Ob  Gr  } $
 +
consists of all groups, the class $  \textrm{ Mor  Gr  } $
 +
of all group homomorphisms, and composition again is composition of homomorphisms (see [[Category of groups|Category of groups]]). One defines in a similar fashion the categories of vector spaces over some field, the category of rings, etc.
  
4) The category of binary relations between sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074057.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074058.png" />). The class of objects of this category is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074059.png" />, but as morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074060.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074061.png" /> one takes all binary relations, that is, all subsets of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074062.png" />; composition is composition of binary relations (cf. [[Binary relation|Binary relation]]).
+
4) The category of binary relations between sets $  \textrm{ Rel  Ens  } $(
 +
or $  R ( \mathfrak S ) $).  
 +
The class of objects of this category is $  \textrm{ Ob  Ens  } $,  
 +
but as morphisms from $  A $
 +
to $  B $
 +
one takes all binary relations, that is, all subsets of the Cartesian product $  A \times B $;  
 +
composition is composition of binary relations (cf. [[Binary relation|Binary relation]]).
  
 
5) A [[monoid]] (semi-group with identity) is a category with a single object; conversely, every category consisting of a single object is a monoid.
 
5) A [[monoid]] (semi-group with identity) is a category with a single object; conversely, every category consisting of a single object is a monoid.
  
6) A pre-ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074063.png" /> can be regarded as a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074064.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074066.png" />, while the product is defined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074067.png" />.
+
6) A [[pre-order]]ed set $(N,{\le})$
 +
can be regarded as a category $  \mathfrak N $
 +
for which $  \mathop{\rm Ob}  \mathfrak N = N $,
 +
$  \mathop{\rm Mor}  \mathfrak N = \{ {( a , b ) } : {a , b \in N, a \leq  b } \} $,  
 +
while the product is defined by the equality $  ( a , b ) ( b , c ) = ( a , c ) $.
  
All categories listed above admit an "isomorphic imbedding" into the category of sets. A category with this property is called a concrete category. Not every category is concrete; for example, the category with as objects all topological spaces and whose morphisms are defined to be homotopy classes of continuous mappings [[#References|[10]]].
+
All categories listed above admit an "isomorphic imbedding" into the category of sets. A category with this property is called a ''concrete category''. Not every category is concrete; for example, the category with as objects all topological spaces and whose morphisms are defined to be homotopy classes of continuous mappings [[#References|[10]]].
  
 
The number of examples of categories can be considerably enlarged by means of various constructions; foremost, by means of functors of categories or categories of diagrams.
 
The number of examples of categories can be considerably enlarged by means of various constructions; foremost, by means of functors of categories or categories of diagrams.
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074068.png" /> between two categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074070.png" /> is called a covariant functor (cf. also [[Functor|Functor]]) if for each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074072.png" />, for each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074073.png" />, its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074074.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074075.png" />, and if, finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074076.png" /> whenever the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074077.png" /> is defined. If the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074078.png" /> constitute a set, then one can construct the category of diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074079.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074080.png" />, the objects of which are all covariant functors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074081.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074082.png" /> and with as morphisms all natural transformations of these functors.
+
A mapping $  F : \mathfrak C \rightarrow \mathfrak C ^ { \prime } $
 +
between two categories $  \mathfrak C $,  
 +
$  \mathfrak C ^ { \prime } $
 +
is called a covariant functor (cf. also [[Functor|Functor]]) if for each object $  A \in  \mathop{\rm Ob}  \mathfrak C $,  
 +
$  F (A) \in  \mathop{\rm Ob}  \mathfrak C ^ { \prime } $,  
 +
for each morphism $  \alpha \in H _ {\mathfrak C }  ( A , B ) $,  
 +
its image $  F ( \alpha ) \in H _ {\mathfrak C ^ { \prime }  } ( F (A) , F (B) ) $,  
 +
if $  F ( 1 _ {A} ) = 1 _ {F (A) }  $,  
 +
and if, finally, $  F ( \alpha \beta ) = F ( \alpha ) F ( \beta ) $
 +
whenever the product $  \alpha \beta $
 +
is defined. If the objects of $  \mathfrak C $
 +
constitute a set, then one can construct the category of diagrams $  \mathop{\rm Funct} ( \mathfrak C , \mathfrak C ^ { \prime } ) $
 +
or $  F ( \mathfrak C , \mathfrak C ^ { \prime } ) $,  
 +
the objects of which are all covariant functors from $  \mathfrak C $
 +
to $  \mathfrak C ^ { \prime } $
 +
and with as morphisms all natural transformations of these functors.
  
To each category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074083.png" /> one can associate its dual category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074084.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074085.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074086.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074088.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074089.png" />. A covariant functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074090.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074091.png" /> is called a contravariant functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074092.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074093.png" />. Along with functors of one argument one can consider many-placed functors or functors of several arguments; cf. [[Functor|Functor]].
+
To each category $  \mathfrak C $
 +
one can associate its dual category $  \mathfrak C $,  
 +
or $  \mathfrak C  ^ {T} $,  
 +
or $  \mathfrak C  ^ {op} $,  
 +
for which $  \mathop{\rm Ob}  \mathfrak C  ^ {*} = \mathop{\rm Ob}  \mathfrak C $
 +
and $  H _ {\mathfrak C  ^ {*}  } ( A , B ) = H _ {\mathfrak C }  ( B , A ) $
 +
for any $  A , B \in  \mathop{\rm Ob}  \mathfrak C $.  
 +
A covariant functor from $  \mathfrak C  ^ {*} $
 +
to $  \mathfrak C ^ { \prime } $
 +
is called a contravariant functor from $  \mathfrak C $
 +
to $  \mathfrak C ^ { \prime } $.  
 +
Along with functors of one argument one can consider many-placed functors or functors of several arguments; cf. [[Functor|Functor]].
  
For each statement in category theory there is a dual statement, obtained by a formal "reversal of the arrows" . In this connection the so-called duality principle holds: A statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074094.png" /> is true in category theory if and only if the dual statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074095.png" /> is true.
+
For each statement in category theory there is a dual statement, obtained by a formal "reversal of the arrows" . In this connection the so-called duality principle holds: A statement $  p $
 +
is true in category theory if and only if the dual statement $  p  ^ {*} $
 +
is true.
  
 
Many concepts and results in mathematics turn out to be dual to others from a category-theoretic point of view: injectivity and projectivity, nilpotency and the notion of category of a topological space in the sense of Lyusternik–Shnirel'man, varieties and radicals in algebra, etc.
 
Many concepts and results in mathematics turn out to be dual to others from a category-theoretic point of view: injectivity and projectivity, nilpotency and the notion of category of a topological space in the sense of Lyusternik–Shnirel'man, varieties and radicals in algebra, etc.
  
A category-theoretic analysis of the foundations of homology theory led to the introduction in the mid-fifties of so-called Abelian categories (cf. [[Abelian category|Abelian category]]). Within this framework it proved possible to realize the basic constructions of homological algebra [[#References|[2]]]. In the 1960s, interest in non-Abelian categories grew, as a result of problems in logic, general algebra, topology, and algebraic geometry. An intensive development of universal algebra and the axiomatic construction of homotopy theory marked the beginnings of various lines of investigation: the category-theoretic study of varieties of universal algebras, the theory of isomorphisms of direct decompositions, the theory of adjoint functors, and the theory of duality of functors. The latter development uncovered the existence of relations between these areas of study. As a result of the recent theory of relative categories, which makes wide use of the techniques of adjoint functors and closed categories, a duality has been established between homotopy theory and the theory of universal algebras; this is based on the interpretation of the categoric definitions of a monoid and a comonoid in suitable categories of functors (see, for example, [[#References|[7]]]). Along with the development of the general theory of relative categories special classes of such categories were introduced: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074096.png" />-categories or formal categories; categories with an involution or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074097.png" />-categories, including, in particular, the category of binary relations; etc. A special case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074098.png" />-categories is the category of small categories, which can be placed at the basis of an axiomatic construction of mathematics.
+
A category-theoretic analysis of the foundations of homology theory led to the introduction in the mid-fifties of so-called Abelian categories (cf. [[Abelian category|Abelian category]]). Within this framework it proved possible to realize the basic constructions of homological algebra [[#References|[2]]]. In the 1960s, interest in non-Abelian categories grew, as a result of problems in logic, general algebra, topology, and algebraic geometry. An intensive development of universal algebra and the axiomatic construction of homotopy theory marked the beginnings of various lines of investigation: the category-theoretic study of varieties of universal algebras, the theory of isomorphisms of direct decompositions, the theory of adjoint functors, and the theory of duality of functors. The latter development uncovered the existence of relations between these areas of study. As a result of the recent theory of relative categories, which makes wide use of the techniques of adjoint functors and closed categories, a duality has been established between homotopy theory and the theory of universal algebras; this is based on the interpretation of the categoric definitions of a monoid and a comonoid in suitable categories of functors (see, for example, [[#References|[7]]]). Along with the development of the general theory of relative categories special classes of such categories were introduced: $  2 $-
 +
categories or formal categories; categories with an involution or $  1 $-
 +
categories, including, in particular, the category of binary relations; etc. A special case of $  2 $-
 +
categories is the category of [[Small category|small categories]], which can be placed at the basis of an axiomatic construction of mathematics.
  
The classes of categories listed above are characterized by the fact that their sets of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c02074099.png" /> possess an additional structure. Another method of introducing an additional structure in a category is to provide the category with a Grothendieck topology and to construct the category of sheaves over the topologized category (so-called topoi; cf. [[Topos|Topos]]).
+
The classes of categories listed above are characterized by the fact that their sets of morphisms $  H ( A , B ) $
 +
possess an additional structure. Another method of introducing an additional structure in a category is to provide the category with a Grothendieck topology and to construct the category of sheaves over the topologized category (so-called topoi; cf. [[Topos|Topos]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) {{MR|0236236}} {{ZBL|0197.29205}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique" ''Tôhoku Math. J.'' , '''9''' (1957) pp. 119–221 {{MR|0102537}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Kurosh, A.Kh. Livshits, E.G. Shul'geifer, "Foundations of category theory" ''Uspekhi Mat. Nauk'' , '''15''' : 6 (1960) pp. 3–52 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> ''Itogi Nauk. Algebra. Topol. 1962'' (1963) pp. 90–106</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> ''Itogi Nauk. Algebra. Topol. Geom. 1967'' (1969) pp. 9–57</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.C. Bunge, "Relative functor categories and categories of algebras" ''J. of Algebra'' , '''11''' (1969) pp. 64–101 {{MR|0236238}} {{ZBL|0165.32902}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, ''Trans. Amer. Math. Soc.'' , '''58''' (1945) pp. 231–294</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper &amp; Row (1964) {{MR|0166240}} {{ZBL|0121.02103}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P. Freyd, "On the concreteness of certain categories" , ''Symp. Math.'' , '''4''' , Acad. Press (1970) pp. 431–456 {{MR|0271184}} {{ZBL|0248.18008}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician" , Springer (1971) {{MR|}} {{ZBL|0232.18001}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Schubert, "Categories" , '''1–2''' , Springer (1972) {{MR|0349793}} {{ZBL|0253.18002}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> B. Mitchell, "Theory of categories" , Acad. Press (1965) {{MR|0202787}} {{ZBL|0136.00604}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) {{MR|0236236}} {{ZBL|0197.29205}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, "Sur quelques points d'algèbre homologique" ''Tôhoku Math. J.'' , '''9''' (1957) pp. 119–221 {{MR|0102537}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Kurosh, A.Kh. Livshits, E.G. Shul'geifer, "Foundations of category theory" ''Uspekhi Mat. Nauk'' , '''15''' : 6 (1960) pp. 3–52 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> ''Itogi Nauk. Algebra. Topol. 1962'' (1963) pp. 90–106</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> ''Itogi Nauk. Algebra. Topol. Geom. 1967'' (1969) pp. 9–57</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.C. Bunge, "Relative functor categories and categories of algebras" ''J. of Algebra'' , '''11''' (1969) pp. 64–101 {{MR|0236238}} {{ZBL|0165.32902}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, ''Trans. Amer. Math. Soc.'' , '''58''' (1945) pp. 231–294</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper &amp; Row (1964) {{MR|0166240}} {{ZBL|0121.02103}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P. Freyd, "On the concreteness of certain categories" , ''Symp. Math.'' , '''4''' , Acad. Press (1970) pp. 431–456 {{MR|0271184}} {{ZBL|0248.18008}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician" , Springer (1971) {{MR|}} {{ZBL|0232.18001}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Schubert, "Categories" , '''1–2''' , Springer (1972) {{MR|0349793}} {{ZBL|0253.18002}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> B. Mitchell, "Theory of categories" , Acad. Press (1965) {{MR|0202787}} {{ZBL|0136.00604}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The notion of a category was introduced in 1942 by S. Eilenberg and S. MacLane [[#References|[a1]]], and has since found numerous applications in algebra, topology and the foundations of mathematics. The intuitive idea is that a category consists of all the objects in some "universe of mathematical discourse" together with all the mappings between them. By identifying an object with its identity morphism it is possible to define the notion of a category in terms of morphisms alone. It is intuitively clearer and also more customary to use both objects and morphisms. The two approaches are mixed-up to some extent in the article above. The object and morphism definition is as follows. A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740100.png" /> consists of
+
The notion of a category was introduced in 1942 by S. Eilenberg and S. MacLane [[#References|[a1]]], and has since found numerous applications in algebra, topology and the foundations of mathematics. The intuitive idea is that a category consists of all the objects in some "universe of mathematical discourse" together with all the mappings between them. By identifying an object with its identity morphism it is possible to define the notion of a category in terms of morphisms alone. It is intuitively clearer and also more customary to use both objects and morphisms. The two approaches are mixed-up to some extent in the article above. The object and morphism definition is as follows. A category $  \mathfrak C $
 +
consists of
  
A1) a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740101.png" /> whose elements are called objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740102.png" />, and a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740103.png" /> whose elements are called morphisms or arrows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740104.png" />;
+
A1) a class $  \mathop{\rm Ob}  \mathfrak C $
 +
whose elements are called objects of $  \mathfrak C $,  
 +
and a class $  \mathop{\rm Mor}  \mathfrak C $
 +
whose elements are called morphisms or arrows of $  \mathfrak C $;
  
A2) operations assigning to each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740106.png" /> a pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740107.png" />, called the domain and codomain (or source and target) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740108.png" />. One writes "a: A B" or "AaB" to mean "a is a morphism with domain A and codomain B" ; this rephrases 1) above;
+
A2) operations assigning to each morphism $  \alpha $
 +
of $  \mathfrak C $
 +
a pair of objects $  ( d _ {0} ( \alpha ) , d _ {1} ( \alpha ) ) $,  
 +
called the domain and codomain (or source and target) of $  \alpha $.  
 +
One writes "a: A B" or "AaB" to mean "a is a morphism with domain A and codomain B" ; this rephrases 1) above;
  
A3) an operation assigning to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740110.png" /> a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740111.png" />, called the identity morphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740112.png" />; this is the precise meaning of part of 5) above;
+
A3) an operation assigning to each object $  A $
 +
of $  \mathfrak C $
 +
a morphism $  1 _ {A} : A \rightarrow A $,  
 +
called the identity morphism on $  A $;  
 +
this is the precise meaning of part of 5) above;
  
A4) a partial (binary) product operation for morphisms, the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740113.png" /> (called the composite of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740115.png" />) being defined if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740116.png" />, and satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740118.png" /> whenever it is defined; this rephrases 3);
+
A4) a partial (binary) product operation for morphisms, the product $  \alpha \beta $(
 +
called the composite of $  \alpha $
 +
and $  \beta $)  
 +
being defined if and only if $  d _ {0} ( \alpha ) = d _ {1} ( \beta ) $,  
 +
and satisfying $  d _ {1} ( \alpha \beta ) = d _ {1} ( \alpha ) $
 +
and  $  d _ {0} ( \alpha \beta ) = d _ {0} ( \beta ) $
 +
whenever it is defined; this rephrases 3);
  
 
these data being subject to the axioms
 
these data being subject to the axioms
  
A5) composition is associative, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740119.png" /> whenever both sides are defined; this rephrases 4) above;
+
A5) composition is associative, i.e. $  ( \alpha \beta ) \gamma = \alpha ( \beta \gamma ) $
 +
whenever both sides are defined; this rephrases 4) above;
  
A6) identity morphisms are units for composition, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740121.png" /> whenever the composites are defined; this and A3) rephrase 5).
+
A6) identity morphisms are units for composition, i.e. $  \alpha 1 _ {A} = \alpha $
 +
and $  1 _ {A} \beta = \beta $
 +
whenever the composites are defined; this and A3) rephrase 5).
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740123.png" /> are not required to be sets, and in many of the leading examples (see, e.g. the main text above and the examples (A1)–(A4), (A7) below) they are not sets. However, most examples have the property that, for each pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740124.png" />, the collection of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740125.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740127.png" /> forms a set (usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740128.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740129.png" />); such categories are sometimes called locally small, although other writers include this condition as part of the definition of a category. A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740130.png" /> is said to be small if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740132.png" /> are sets (cf. [[Small category|Small category]]); it turns out that many of the fundamental mathematical structures may be regarded as small categories (e.g. (A6) and (A7) below).
+
The classes $  \mathop{\rm Ob}  \mathfrak C $
 +
and $  \mathop{\rm Mor}  \mathfrak C $
 +
are not required to be sets, and in many of the leading examples (see, e.g. the main text above and the examples (A1)–(A4), (A7) below) they are not sets. However, most examples have the property that, for each pair of objects $  ( A , B ) $,  
 +
the collection of morphisms $  \alpha $
 +
with $  d _ {0} ( \alpha ) = A $
 +
and $  d _ {1} ( \alpha ) = B $
 +
forms a set (usually denoted by $  \mathfrak C ( A , B ) $
 +
or $  \mathop{\rm Hom} ( A , B ) $);  
 +
such categories are sometimes called locally small, although other writers include this condition as part of the definition of a category. A category $  \mathfrak C $
 +
is said to be small if $  \mathop{\rm Ob}  \mathfrak C $
 +
and $  \mathop{\rm Mor}  \mathfrak C $
 +
are sets (cf. [[Small category]]); it turns out that many of the fundamental mathematical structures may be regarded as small categories (e.g. (A6) and (A7) below).
  
 
It follows from the definition that each object in a category has a unique identity morphism; thus it is possible to identify objects with their identity morphisms, leading to an axiomatization of categories in which "morphism" and "composite" are the only primitive notions (see [[#References|[9]]]).
 
It follows from the definition that each object in a category has a unique identity morphism; thus it is possible to identify objects with their identity morphisms, leading to an axiomatization of categories in which "morphism" and "composite" are the only primitive notions (see [[#References|[9]]]).
Line 80: Line 217:
 
Examples of categories.
 
Examples of categories.
  
(A1) See the main article. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740133.png" /> of sets is more often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740134.png" />.
+
(A1) See the main article. The category $  \mathop{\rm Ens} $
 +
of sets is more often denoted by $  \mathop{\rm Set} $.
 +
 
 +
(A3) Next to the category  $  \mathop{\rm Gr} $
 +
one may consider the categories  $  \mathop{\rm Ab} $
 +
of Abelian groups,  $  \mathop{\rm Mod} _ {R} $
 +
of (right) modules over a fixed ring  $  R $,
 +
etc.
 +
 
 +
(A4) The category of binary relations between sets is usually denoted by  $  \mathop{\rm Rel} $.
  
(A3) Next to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740135.png" /> one may consider the categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740136.png" /> of Abelian groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740137.png" /> of (right) modules over a fixed ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740138.png" />, etc.
+
(A5) A semi-group with identity  $  M $
 +
is also called a [[Monoid|monoid]]. It defines a category with one object  $  * $,
 +
the elements of $  M $
 +
being interpreted as morphisms  $  * \rightarrow * $.
  
(A4) The category of binary relations between sets is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740139.png" />.
+
(A6) A [[Partially ordered set|partially ordered set]]  $  ( A , \leq  ) $
 +
defines a category whose objects are the elements of $  A $,
 +
and whose morphisms are the instances of the order-relation: that is, there is just one morphism  $  a \rightarrow b $
 +
if  $  a \leq  b $,
 +
and none otherwise.
  
(A5) A semi-group with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740140.png" /> is also called a [[Monoid|monoid]]. It defines a category with one object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740141.png" />, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740142.png" /> being interpreted as morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740143.png" />.
+
(A7) Similarly, an oriented [[Graph|graph]] (or diagram scheme) can be interpreted as a category of which the objects are the vertices and the morphism from  $  v _ {1} $
 +
to  $  v _ {2} $
 +
are the oriented paths from  $  v _ {1} $
 +
to  $  v _ {2} $
 +
including the trivial identity paths from  $  v $
 +
to  $  v $
 +
for all vertices. Inversely a category can be seen as a (very large) oriented graph with loops and multiple edges together with an equivalence relation identifying certain paths, cf. [[#References|[11]]], Section II.7.
  
(A6) A [[Partially ordered set|partially ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740144.png" /> defines a category whose objects are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740145.png" />, and whose morphisms are the instances of the order-relation: that is, there is just one morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740146.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740147.png" />, and none otherwise.
+
(A8) The homotopy category $  \mathop{\rm Htpy} $
 +
or  $  \mathop{\rm Htp} $
 +
has the same objects as  $  \mathop{\rm Top} $,  
 +
but its morphisms are homotopy classes of continuous mappings. This category can be proved to be nonconcrete.
  
(A7) Similarly, an oriented [[Graph|graph]] (or diagram scheme) can be interpreted as a category of which the objects are the vertices and the morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740148.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740149.png" /> are the oriented paths from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740150.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740151.png" /> including the trivial identity paths from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740152.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740153.png" /> for all vertices. Inversely a category can be seen as a (very large) oriented graph with loops and multiple edges together with an equivalence relation identifying certain paths, cf. [[#References|[11]]], Section II.7.
+
In the study of categories, functors (morphisms of categories) play an essential role. A [[Functor|functor]] $  F :  \mathfrak C \rightarrow \mathfrak D $
 +
consists of two functions, one assigning to each object  $  A $
 +
of  $  \mathfrak C $
 +
an object  $  FA $
 +
of $  \mathfrak D $,
 +
and the other assigning to each morphism  $  \alpha $
 +
of  $  \mathfrak C $
 +
a morphism $  F \alpha $
 +
of  $  \mathfrak D $,
 +
in such a way that the categorical structure is preserved: $  d _ {i} ( F \alpha ) = F ( d _ {i} ( \alpha ) ) $
 +
$  ( i = 0 , 1 ) $,
 +
$  F ( 1 _ {A} ) = 1 _ {FA} $
 +
and  $  F ( \alpha \beta ) = ( F \alpha ) ( F \beta ) $
 +
whenever  $  \alpha \beta $
 +
is defined. There is also a third level of structure: if  $  F $
 +
and  $  G $
 +
are both functors  $  \mathfrak C \rightarrow \mathfrak D $,
 +
a natural transformation (or functorial morphism  $  \eta : F \rightarrow G $
 +
is a function assigning to each object  $  A $
 +
of  $  \mathfrak C $
 +
a morphism  $  \eta _ {A} : F A \rightarrow G A $
 +
in  $  \mathfrak D $,
 +
such that for every  $  A \rightarrow  ^  \alpha  B $
 +
in  $  \mathfrak C $
 +
the diagram
  
(A8) The homotopy category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740154.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740155.png" /> has the same objects as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740156.png" />, but its morphisms are homotopy classes of continuous mappings. This category can be proved to be nonconcrete.
+
$$
  
In the study of categories, functors (morphisms of categories) play an essential role. A [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740157.png" /> consists of two functions, one assigning to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740158.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740159.png" /> an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740160.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740161.png" />, and the other assigning to each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740162.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740163.png" /> a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740164.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740165.png" />, in such a way that the categorical structure is preserved: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740166.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740169.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740170.png" /> is defined. There is also a third level of structure: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740171.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740172.png" /> are both functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740173.png" />, a natural transformation (or functorial morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740174.png" /> is a function assigning to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740175.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740176.png" /> a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740177.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740178.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740179.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740180.png" /> the diagram
+
\begin{array}{rcl}
 +
FA  & \mathop \rightarrow \limits ^ { {F \alpha }}  &FB  \\
 +
\eta _ {A} \downarrow  &{}  &\downarrow \eta _ {B}  \\
 +
GA  & \mathop \rightarrow \limits _ { {G \alpha }}  &GB  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740181.png" /></td> </tr></table>
+
$$
  
commutes (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740182.png" />). Functors may be composed (and every category has an identity functor); thus there is a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740183.png" /> of (small) categories and functors between them. Natural transformations may be composed; thus, given two categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740184.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740185.png" />, there is a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740186.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740187.png" />) of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740188.png" /> and natural transformations between them. This is one important way in which new categories are constructed from existing ones. The resulting categories are called categories of functors or categories of diagrams. The latter name is especially understandable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740189.png" /> is the category corresponding to a diagram scheme (oriented graph). Indeed, then a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740190.png" /> "is" a diagram in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740191.png" />. Other important constructions to obtain new categories are: taking quotients (cf. [[Quotient category|Quotient category]]), taking localizations (cf. [[Localization in categories|Localization in categories]]) and constructing relative and comma categories. The important notion of a [[Derived category|derived category]] involves several of these constructions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740192.png" /> is a category and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740193.png" /> an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740194.png" />, then the relative category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740195.png" /> of objects over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740196.png" /> has as objects all morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740197.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740198.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740199.png" /> and a morphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740200.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740201.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740202.png" /> is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740203.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740204.png" />. Dually there is the notion of the relative category of objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740205.png" /> under a given object. Intuitively an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740207.png" /> is a family of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740208.png" /> parametrized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740209.png" /> or a fibre object (fibred object). The systematic consideration of these relative objects, i.e. fibre objects (and their duals) combined with [[Base change|base change]] and [[Deformation|deformation]] ideas has become a most important technique in many parts of mathematics, especially in algebra (notably homological algebra), algebraic and differential geometry, topology, and differential and algebraic topology. It is especially important to find the right fibrewise versions of definitions, theorems and concepts. (A second additional not unrelated major trend involves finding the right equivariant versions in those case in which there is a group of symmetries present (as well).)
+
commutes (i.e. $  ( G \alpha ) ( \eta _ {A} ) = ( \eta _ {B} ) ( F \alpha ) $).  
 +
Functors may be composed (and every category has an identity functor); thus there is a category $  Cat $
 +
of (small) categories and functors between them. Natural transformations may be composed; thus, given two categories $  \mathfrak C $
 +
and $  \mathfrak D $,  
 +
there is a category $  [ \mathfrak C , \mathfrak D ] $(
 +
or $  \mathfrak D ^ {\mathfrak C } $)  
 +
of functors $  \mathfrak C \rightarrow \mathfrak D $
 +
and natural transformations between them. This is one important way in which new categories are constructed from existing ones. The resulting categories are called categories of functors or categories of diagrams. The latter name is especially understandable if $  \mathfrak C $
 +
is the category corresponding to a diagram scheme (oriented graph). Indeed, then a functor $  \mathfrak C \rightarrow \mathfrak D $"
 +
is" a diagram in $  \mathfrak D $.  
 +
Other important constructions to obtain new categories are: taking quotients (cf. [[Quotient category|Quotient category]]), taking localizations (cf. [[Localization in categories|Localization in categories]]) and constructing relative and comma categories. The important notion of a [[Derived category|derived category]] involves several of these constructions. If $  \mathfrak C $
 +
is a category and $  B $
 +
an object of $  \mathfrak C $,  
 +
then the relative category $  \mathfrak C / B $
 +
of objects over $  B $
 +
has as objects all morphisms $  \alpha : A \rightarrow B $
 +
of $  \mathfrak C $
 +
into $  B $
 +
and a morphism in $  \mathfrak C / \mathfrak B $
 +
of $  \alpha $
 +
to $  \alpha  ^  \prime  : A ^ { \prime } \rightarrow B $
 +
is a morphism $  \phi : A \rightarrow A ^ { \prime } $
 +
such that $  \alpha  ^  \prime  \circ \phi = \alpha $.  
 +
Dually there is the notion of the relative category of objects in $  \mathfrak C $
 +
under a given object. Intuitively an object $  \alpha : A \rightarrow B $
 +
of $  \mathfrak C / B $
 +
is a family of objects of $  \mathfrak C $
 +
parametrized by $  B $
 +
or a fibre object (fibred object). The systematic consideration of these relative objects, i.e. fibre objects (and their duals) combined with [[Base change|base change]] and [[Deformation|deformation]] ideas has become a most important technique in many parts of mathematics, especially in algebra (notably homological algebra), algebraic and differential geometry, topology, and differential and algebraic topology. It is especially important to find the right fibrewise versions of definitions, theorems and concepts. (A second additional not unrelated major trend involves finding the right equivariant versions in those case in which there is a group of symmetries present (as well).)
  
The idea of a comma category generalizes that of categories of objects over or under a given object. Let there be given three categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740210.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740212.png" /> and two functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740213.png" /> arranged as follows
+
The idea of a comma category generalizes that of categories of objects over or under a given object. Let there be given three categories $  \mathfrak C $,  
 +
$  \mathfrak B $,  
 +
$  \mathfrak A $
 +
and two functors $  S , T $
 +
arranged as follows
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740214.png" /></td> </tr></table>
+
$$
 +
\mathfrak A  \rightarrow ^ { T }  \mathfrak C  \leftarrow ^ { S }  \mathfrak B .
 +
$$
  
Then the comma category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740215.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740216.png" /> has as objects all triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740217.png" /> consisting of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740218.png" />, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740219.png" /> and a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740220.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740221.png" />. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740222.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740223.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740224.png" /> consists of a pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740225.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740226.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740227.png" />.
+
Then the comma category $  ( T , S ) $
 +
or $  ( T \downarrow S ) $
 +
has as objects all triples $  ( a , f , b ) $
 +
consisting of an object $  a \in \mathfrak A $,  
 +
an object $  b \in \mathfrak B $
 +
and a morphism $  f : T A \rightarrow S B $
 +
in $  \mathfrak C $.  
 +
A morphism $  ( \alpha , \beta ) $
 +
from $  ( a , f , b ) $
 +
to $  ( a  ^  \prime  , f ^ { \prime } , b  ^  \prime  ) $
 +
consists of a pair of morphisms $  \alpha : A \rightarrow A ^ { \prime } $
 +
and $  \beta : B \rightarrow B ^ { \prime } $
 +
such that $  f ^ { \prime } \circ T \alpha = T \beta \circ f $.
  
Examples of functors. ITEM {(A9} The forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740228.png" /> sends each topological space to its underlying set, and each continuous function to itself ( "forgetting" the continuity). Similarly, one has forgetful functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740229.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740230.png" />, etc.
+
Examples of functors. ITEM {(A9} The forgetful functor $  \mathop{\rm Top} \rightarrow  \mathop{\rm Set} $
 +
sends each topological space to its underlying set, and each continuous function to itself ( "forgetting" the continuity). Similarly, one has forgetful functors $  \mathop{\rm Gr} \rightarrow  \mathop{\rm Set} $,  
 +
$  \mathop{\rm Mod} _ {R} \rightarrow  \mathop{\rm Ab} $,  
 +
etc.
  
(A10) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740231.png" /> is a locally small category, then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740232.png" /> there is a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740233.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740234.png" /> to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740235.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740236.png" /> to the function which sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740237.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740238.png" />. Such functors (or functors isomorphic to them in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740239.png" />) are called representable (cf. [[Representable functor|Representable functor]]).
+
(A10) If $  \mathfrak C $
 +
is a [[locally small category]], then for each $  A \in  \mathop{\rm Ob}  \mathfrak C $
 +
there is a functor $  \mathfrak C ( A , - ) : \mathfrak C \rightarrow  \mathop{\rm Set} $
 +
sending $  B $
 +
to the set $  \mathfrak C ( A , B ) $
 +
and $  B \rightarrow  ^  \alpha  B ^ { \prime } $
 +
to the function which sends $  \beta $
 +
to $  \alpha \beta $.  
 +
Such functors (or functors isomorphic to them in $  [ \mathfrak C ,  \mathop{\rm Set} ] $)  
 +
are called representable (cf. [[Representable functor|Representable functor]]).
  
(A11) There is a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740240.png" /> sending a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740241.png" /> to the free group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740242.png" />, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740243.png" /> to the unique homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740244.png" /> sending the generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740245.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740246.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740247.png" />, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740248.png" />.
+
(A11) There is a functor $  F :   \mathop{\rm Set} \rightarrow  \mathop{\rm Gr} $
 +
sending a set $  A $
 +
to the free group generated by $  A $,  
 +
and a function $  A \rightarrow  ^  \alpha  B $
 +
to the unique homomorphism $  F A \rightarrow F B $
 +
sending the generator $  a $
 +
of $  F A $
 +
to $  \alpha (a) \in F B $,  
 +
for each $  a \in A $.
  
(A12) The (singular) homology groups of spaces (cf. [[Homology group|Homology group]]) define functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740249.png" /> (one for each dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740250.png" />).
+
(A12) The (singular) homology groups of spaces (cf. [[Homology group|Homology group]]) define functors $  \mathop{\rm Htpy} \rightarrow  \mathop{\rm AB} $(
 +
one for each dimension $  \geq  0 $).
  
 
(A13) A functor between monoids, considered as categories, is just a monoid homomorphism.
 
(A13) A functor between monoids, considered as categories, is just a monoid homomorphism.
Line 118: Line 375:
 
(A14) A functor between partially ordered sets, considered as categories, is just an order-preserving mapping.
 
(A14) A functor between partially ordered sets, considered as categories, is just an order-preserving mapping.
  
(A15) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740251.png" /> is a [[Group|group]], considered as a category, a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740252.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740253.png" />) is a permutation (respectively, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740254.png" />-linear representation) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740255.png" /> (cf. [[Representation of a group|Representation of a group]]).
+
(A15) If $  G $
 +
is a [[Group|group]], considered as a category, a functor $  G \rightarrow  \mathop{\rm Set} $(
 +
respectively, $  G \rightarrow  \mathop{\rm Mod} _ {R} $)  
 +
is a permutation (respectively, an $  R $-
 +
linear representation) of $  G $(
 +
cf. [[Representation of a group|Representation of a group]]).
  
A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740256.png" /> is said to be faithful (cf. [[Faithful functor|Faithful functor]]) if it is "injective on Hom-sets" ; i.e. if, given two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740257.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740258.png" /> with the same domain and codomain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740259.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740260.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740261.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740262.png" /> is said to be full if it is "surjective on Hom-sets" in a similar sense. Forgetful functors (as in (A9) above) are always faithful. The property that a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740263.png" /> be concrete can now be rephrased as: There is a faithful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740264.png" />.
+
A functor $  F : \mathfrak C \rightarrow \mathfrak D $
 +
is said to be faithful (cf. [[Faithful functor|Faithful functor]]) if it is "injective on Hom-sets" ; i.e. if, given two morphisms $  A \rightarrow  ^  \alpha  B $
 +
and $  A \rightarrow  ^  \beta  B $
 +
with the same domain and codomain in $  \mathfrak C $,  
 +
$  F \alpha = F \beta $
 +
implies $  \alpha = \beta $.  
 +
$  F $
 +
is said to be full if it is "surjective on Hom-sets" in a similar sense. Forgetful functors (as in (A9) above) are always faithful. The property that a category $  \mathfrak C $
 +
be concrete can now be rephrased as: There is a faithful functor $  \mathfrak C \rightarrow  \mathop{\rm Set} $.
  
Given a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740265.png" />, one can form its opposite or [[Dual category|dual category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740266.png" /> by keeping the same objects as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740267.png" /> and reversing all the morphisms. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740268.png" /> is isomorphic to its opposite, though most familiar categories are not. A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740269.png" /> is sometimes called a contravariant functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740270.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740271.png" />; for emphasis, functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740272.png" /> are then called covariant. (For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740273.png" /> is locally small, one may define a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740274.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740275.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740276.png" />, by analogy with the covariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740277.png" /> of example (A10) above.) The [[Duality principle|duality principle]] for categories is essentially the assertion that something which is true for all categories is true for the duals of all categories.
+
Given a category $  \mathfrak C $,  
 +
one can form its opposite or [[Dual category|dual category]] $  \mathfrak C  ^ {op} $
 +
by keeping the same objects as $  \mathfrak C $
 +
and reversing all the morphisms. The category $  \mathop{\rm Rel} $
 +
is isomorphic to its opposite, though most familiar categories are not. A functor $  \mathfrak C  ^ {op} \rightarrow \mathfrak D $
 +
is sometimes called a contravariant functor from $  \mathfrak C $
 +
to $  \mathfrak D $;  
 +
for emphasis, functors $  \mathfrak C \rightarrow \mathfrak D $
 +
are then called covariant. (For example, if $  \mathfrak C $
 +
is locally small, one may define a contravariant functor $  \mathfrak C ( - , A ) $
 +
from $  \mathfrak C $
 +
to $  \mathop{\rm Set} $,  
 +
by analogy with the covariant functor $  \mathfrak C ( A , - ) $
 +
of example (A10) above.) The [[Duality principle|duality principle]] for categories is essentially the assertion that something which is true for all categories is true for the duals of all categories.
  
 
S. MacLane [[#References|[a4]]] introduced the idea that Cartesian products can be characterized in categorical terms, by a universal property; this gave rise to the general categorical notion of limit (and the dual notion of colimit), which includes products as a special case (cf. [[Limit|Limit]] and [[Universal problems|Universal problems]]).
 
S. MacLane [[#References|[a4]]] introduced the idea that Cartesian products can be characterized in categorical terms, by a universal property; this gave rise to the general categorical notion of limit (and the dual notion of colimit), which includes products as a special case (cf. [[Limit|Limit]] and [[Universal problems|Universal problems]]).
  
The key notion of adjunction came latter [[#References|[a5]]]: given functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740278.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740279.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740280.png" /> is left adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740281.png" /> (written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740282.png" />) if there is a bijection between morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740283.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740284.png" /> which is natural in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740285.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740286.png" />; this is equivalent to the existence of natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740287.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740288.png" /> satisfying certain identities [[#References|[11]]] (cf. [[Adjoint functor|Adjoint functor]]). For example, the free group functor (example (A11) above) is left adjoint to the forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740289.png" />; Galois connections (cf. [[Galois correspondence|Galois correspondence]]) are examples of (contravariant) adjunctions between partially ordered sets. A functor which has a left adjoint preserves all limits; the converse implication is valid under suitable "smallness conditions" (the adjoint functor theorem, see [[#References|[9]]]).
+
The key notion of adjunction came latter [[#References|[a5]]]: given functors $  F : \mathfrak C \rightarrow \mathfrak D $
 +
and $  G : \mathfrak D \rightarrow \mathfrak C $,  
 +
one says that $  F $
 +
is left adjoint to $  G $(
 +
written $  F \dashv G $)  
 +
if there is a bijection between morphisms $  F A \rightarrow B $
 +
and morphisms $  A \rightarrow G B $
 +
which is natural in $  A $
 +
and $  B $;  
 +
this is equivalent to the existence of natural transformations $  \eta : 1 _ {\mathfrak C }  \rightarrow G F $
 +
and $  \epsilon : F G \rightarrow 1 _ {\mathfrak D }  $
 +
satisfying certain identities [[#References|[11]]] (cf. [[Adjoint functor|Adjoint functor]]). For example, the free group functor (example (A11) above) is left adjoint to the forgetful functor $  \mathop{\rm Gr} \rightarrow  \mathop{\rm Set} $;  
 +
Galois connections (cf. [[Galois correspondence|Galois correspondence]]) are examples of (contravariant) adjunctions between partially ordered sets. A functor which has a left adjoint preserves all limits; the converse implication is valid under suitable "smallness conditions" (the adjoint functor theorem, see [[#References|[9]]]).
  
Given an adjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740290.png" /> as above, the composite functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740291.png" /> is equipped with natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740292.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740293.png" /> satisfying certain identities; these data define the notion of a monad or [[Triple|triple]] on a category, which played a central role in much categorical research in the 1960's and later years.
+
Given an adjunction $  ( F \dashv G ) $
 +
as above, the composite functor $  T = G F : \mathfrak C \rightarrow \mathfrak C $
 +
is equipped with natural transformations $  \eta : 1 _ {\mathfrak C }  \rightarrow T $
 +
and $  \mu = G \epsilon _ {F} : T T \rightarrow T $
 +
satisfying certain identities; these data define the notion of a monad or [[Triple|triple]] on a category, which played a central role in much categorical research in the 1960's and later years.
  
The identities which a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740294.png" /> on a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740295.png" /> is required to satisfy are the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740296.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740297.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740298.png" />. An algebra for the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740299.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740301.png" />-algebra, is an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740302.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740303.png" /> together with a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740304.png" /> such that the following identities hold: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740306.png" />. It is a good idea to write out these requirements in terms of commutative diagrams. They are reminiscent of associativity and unit requirements.
+
The identities which a triple $  ( T , \mu , \eta ) $
 +
on a category $  \mathfrak C $
 +
is required to satisfy are the following: $  \mu _ {A} \circ T ( \mu _ {A} ) = \mu _ {A} \circ \mu _ {T(A)} $,  
 +
$  \mu _ {A} \circ T ( \eta _ {A} ) = 1 _ {T (A) }  $,  
 +
$  \mu _ {A} \circ \eta _ {T (A) }  = 1 _ {T (A) }  $.  
 +
An algebra for the triple $  T $,  
 +
or $  T $-
 +
algebra, is an object $  X $
 +
of $  \mathfrak C $
 +
together with a morphism $  \xi : T X \rightarrow X $
 +
such that the following identities hold: $  \xi \circ \eta _ {X} = 1 _ {X} $,  
 +
$  \xi \circ T ( \xi ) = \xi \circ \mu _ {X} $.  
 +
It is a good idea to write out these requirements in terms of commutative diagrams. They are reminiscent of associativity and unit requirements.
  
Dually, i.e. reversing all arrows, there is the notion of a cotriple and the corresponding notion of a co-algebra over such a cotriple. An important example of a cotriple in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740307.png" /> of commutative rings with unit element is the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740308.png" /> of the big Witt vectors together with the structure of a special <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740310.png" />-ring on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740311.png" />. The co-algebras for this cotriple are precisely the special <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740312.png" />-algebras (cf. [[Witt vector|Witt vector]] and [[Lambda-ring|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740313.png" />-ring]]).
+
Dually, i.e. reversing all arrows, there is the notion of a cotriple and the corresponding notion of a co-algebra over such a cotriple. An important example of a cotriple in the category $  \mathop{\rm Ring} $
 +
of commutative rings with unit element is the functor $  W : \mathop{\rm Ring} \rightarrow  \mathop{\rm Ring} $
 +
of the big Witt vectors together with the structure of a special $  \lambda $-
 +
ring on $  W (R) $.  
 +
The co-algebras for this cotriple are precisely the special $  \lambda $-
 +
algebras (cf. [[Witt vector|Witt vector]] and [[Lambda-ring| $  \lambda $-
 +
ring]]).
  
Important examples of triples arise from adjunctions involving forgetful functors. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740314.png" /> be the forgetful functor from the category of commutative rings with unit element to the category of sets. This one has an adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740315.png" /> which assigns to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740316.png" /> the free commutative ring with generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740317.png" />, i.e. the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740318.png" /> of commutative polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740319.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740320.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740321.png" />. The freeness property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740322.png" />, i.e. the property that for every ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740323.png" /> and every collection of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740324.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740325.png" /> there is precisely one homomorphism of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740326.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740327.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740328.png" />, precisely expresses the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740329.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740330.png" /> are adjoint functors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740331.png" />. The corresponding natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740332.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740333.png" /> (cf. also [[Adjoint functor|Adjoint functor]]).
+
Important examples of triples arise from adjunctions involving forgetful functors. For example, let $  V :   \mathop{\rm Ring} \rightarrow  \mathop{\rm Set} $
 +
be the forgetful functor from the category of commutative rings with unit element to the category of sets. This one has an adjoint $  F :   \mathop{\rm Set} \rightarrow  \mathop{\rm Ring} $
 +
which assigns to a set $  E \in  \mathop{\rm Set} $
 +
the free commutative ring with generator $  E $,  
 +
i.e. the ring $  \mathbf Z [ X _ {e} : e \in E ] $
 +
of commutative polynomials over $  \mathbf Z $
 +
in the variables $  X _ {e} $,  
 +
$  e \in E $.  
 +
The freeness property of $  \mathbf Z [ X _ {e} : e \in E ] $,  
 +
i.e. the property that for every ring $  A $
 +
and every collection of elements $  ( a _ {e} ) _ {e \in E }  $
 +
of $  A $
 +
there is precisely one homomorphism of rings $  \phi :  \mathbf Z [ X _ {e} : e \in E ] \rightarrow A $
 +
such that $  \phi ( X _ {e} ) = a _ {e} $
 +
for all $  e \in E $,  
 +
precisely expresses the fact that $  V $
 +
and $  F $
 +
are adjoint functors: $  \mathop{\rm Set} ( E , V (A) ) \cong  \mathop{\rm Ring} ( F E , A ) $.  
 +
The corresponding natural transformation $  \mathop{\rm id} \rightarrow V F $
 +
is given by $  e \mapsto X _ {e} $(
 +
cf. also [[Adjoint functor|Adjoint functor]]).
  
Every monad and comonad can be induced by an adjunction; in fact there is a "best possible" such adjunction, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740334.png" /> is taken to be the category of (Eilenberg–Moore) algebras for the monad, [[#References|[a7]]]. A general adjunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740335.png" /> is said to be monadic (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740336.png" /> is said to be monadic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740337.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740338.png" /> is (canonically) equivalent to the category of algebras for the induced monad on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740339.png" />. The adjunction between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740340.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740341.png" />, mentioned above, is monadic; more generally, the categories which are monadic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740342.png" /> can be characterized [[#References|[a8]]] as those which arise from varieties of universal algebras (provided one allows infinitary as well as finitary algebraic operations; the finitary case can also be characterized in categorical terms [[#References|[9]]], using the notion of algebraic theory). See also [[Variety of universal algebras|Variety of universal algebras]].
+
Every monad and comonad can be induced by an adjunction; in fact there is a "best possible" such adjunction, in which $  \mathfrak D $
 +
is taken to be the category of (Eilenberg–Moore) algebras for the monad, [[#References|[a7]]]. A general adjunction $  (F \dashv G ) $
 +
is said to be monadic (or $  \mathfrak D $
 +
is said to be monadic over $  \mathfrak C  $)  
 +
if $  \mathfrak D $
 +
is (canonically) equivalent to the category of algebras for the induced monad on $  \mathfrak C $.  
 +
The adjunction between the $  \mathop{\rm Set} $
 +
and $  \mathop{\rm Gr} $,  
 +
mentioned above, is monadic; more generally, the categories which are monadic over $  \mathop{\rm Set} $
 +
can be characterized [[#References|[a8]]] as those which arise from varieties of universal algebras (provided one allows infinitary as well as finitary algebraic operations; the finitary case can also be characterized in categorical terms [[#References|[9]]], using the notion of algebraic theory). See also [[Variety of universal algebras|Variety of universal algebras]].
  
Another phrase that is used to denote a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740343.png" /> is algebraic theory (in monad form) over the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740344.png" />. It is so to speak the theory of the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740345.png" />-algebras. There are, at least, two more equivalent ways in which this notion is approached. One is as follows [[#References|[a6]]]. An algebraic theory in clone form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740346.png" /> consists of an "object assignment function" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740347.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740348.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740349.png" />-terms with variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740350.png" />) for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740351.png" />, an "insertion of variables mapping" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740352.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740353.png" /> and a "clone-composition function" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740354.png" /> for each ordered triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740355.png" /> of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740356.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740357.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740358.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740359.png" /> be the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740360.png" />. Then the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740361.png" /> are supposed to satisfy the following axioms. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740362.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740363.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740364.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740365.png" />,
+
Another phrase that is used to denote a triple $  ( T , \mu , \eta ) $
 +
is algebraic theory (in monad form) over the category $  \mathfrak C $.  
 +
It is so to speak the theory of the category of $  T $-
 +
algebras. There are, at least, two more equivalent ways in which this notion is approached. One is as follows [[#References|[a6]]]. An algebraic theory in clone form $  ( T , \eta , \circ ) $
 +
consists of an "object assignment function" $  A \mapsto T A $(
 +
= $
 +
$  T $-
 +
terms with variables in $  A $)  
 +
for all objects $  A $,  
 +
an "insertion of variables mapping" $  \eta _ {A} : A \rightarrow T A $
 +
for all $  A $
 +
and a "clone-composition function" $  \mathfrak C ( B , T C ) \times \mathfrak C ( A , T B ) \rightarrow  ^  \circ  \mathfrak C ( A , T C ) $
 +
for each ordered triple $  ( A , B , C ) $
 +
of objects of $  \mathfrak C $.  
 +
For each $  f : A \rightarrow B $
 +
in $  \mathfrak C $
 +
let $  f ^ { \# } $
 +
be the composite $  A \rightarrow B \rightarrow ^ {\eta _ {B} } T B $.  
 +
Then the data $  ( T , \eta , \circ ) $
 +
are supposed to satisfy the following axioms. For all $  \alpha : A \rightarrow T B $,  
 +
$  \beta : B \rightarrow T C $,  
 +
$  \gamma : C \rightarrow T D $
 +
and $  f : A \rightarrow B $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740366.png" /></td> </tr></table>
+
$$
 +
( \gamma \circ \beta ) \circ \alpha  = \gamma \circ ( \beta \circ \alpha ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740367.png" /></td> </tr></table>
+
$$
 +
\eta _ {B} \circ \alpha  = \alpha ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740368.png" /></td> </tr></table>
+
$$
 +
\alpha \circ f ^ { \# }  = \alpha f .
 +
$$
  
This defines a new category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740369.png" />, the Kleishi category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740370.png" />. The objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740371.png" /> are the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740372.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740373.png" />, composition is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740374.png" />, and the identity morphisms are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740375.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740376.png" />.
+
This defines a new category $  \mathfrak C _ {T} $,  
 +
the Kleishi category of $  ( T , \eta , \circ ) $.  
 +
The objects of $  \mathfrak C _ {T} $
 +
are the objects of $  \mathfrak C $,
 +
$  \mathfrak C _ {T} ( A , B ) = \mathfrak C ( A , T B ) $,  
 +
composition is given by $  \circ $,  
 +
and the identity morphisms are the $  \eta _ {A} $
 +
in $  \mathfrak C _ {T} ( A , A ) = \mathfrak C ( A , T A ) $.
  
A simple example of an algebraic theory in clone form is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740377.png" /> be a ring with unit. For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740378.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740379.png" /> be the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740380.png" />. A matrix with columns indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740381.png" /> and rows indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740382.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740383.png" />, i.e. a morphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740384.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740385.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740386.png" />-th column of the matrix. Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740387.png" /> matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740388.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740389.png" /> matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740390.png" />, define their composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740391.png" /> by the usual matrix product, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740392.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740393.png" />-vector with components
+
A simple example of an algebraic theory in clone form is as follows. Let $  R $
 +
be a ring with unit. For a set $  A $
 +
let $  T A $
 +
be the vector space $  R  ^ {(A)} = \{ {( r _ {a} ) _ {a \in A }  } : {r _ {a} \in R  \textrm{ and  only  finitely  many  }  r _ {a } \textrm{ are  different  "_" zero  } } \} $.  
 +
A matrix with columns indexed by $  B $
 +
and rows indexed by $  A $
 +
is a mapping $  \alpha : B \rightarrow T A $,  
 +
i.e. a morphism in $  \mathop{\rm Set} $;  
 +
$  \alpha (b) $
 +
is the $  b $-
 +
th column of the matrix. Given an $  A \times B $
 +
matrix $  \alpha : B \rightarrow T A $
 +
and a $  B \times C $
 +
matrix $  \beta : C \rightarrow T B $,  
 +
define their composite $  \alpha \circ \beta : C \rightarrow T A $
 +
by the usual matrix product, i.e. $  ( \alpha \circ \beta ) (c) $
 +
is the $  A $-
 +
vector with components
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740394.png" /></td> </tr></table>
+
$$
 +
( \alpha \circ \beta ) (c) _ {a}  = \
 +
\sum _ { b } \alpha (b) _ {a} \beta (c) _ {b} .
 +
$$
  
The insertion of variables assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740395.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740396.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740397.png" /> denotes the Kronecker delta (cf. [[Kronecker symbol|Kronecker symbol]]). It is easily checked that the axioms above are satisfied.
+
The insertion of variables assignment $  \eta : A \rightarrow T A $
 +
is defined by $  \eta (a) _ {a  ^  \prime  } = \delta _ {a , a  ^  \prime  } $
 +
where $  \delta $
 +
denotes the Kronecker delta (cf. [[Kronecker symbol|Kronecker symbol]]). It is easily checked that the axioms above are satisfied.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740398.png" /> be an algebraic theory in clone form. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740399.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740400.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740401.png" /> as the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740402.png" />. It follows readily that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740403.png" /> is then a functor and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740404.png" /> is a natural transformation. Further define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740405.png" /> as the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740406.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740407.png" /> is also a natural transformation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740408.png" /> is a triple. Moreover, this construction yielding a triple for each algebraic theory in clone form is a bijection. For a discussion of the algebraic theories (in clone and monad form) coming from a universal algebra and a third categorical way of viewing universal algebras see [[Universal algebra|Universal algebra]].
+
Let $  ( T , \eta , \circ ) $
 +
be an algebraic theory in clone form. For $  f : A \rightarrow B $
 +
in $  \mathfrak C $
 +
define $  T f : T A \rightarrow T B $
 +
as the composite $  f ^ { \# } \circ  \mathop{\rm id} _ {TA} $.  
 +
It follows readily that $  T $
 +
is then a functor and that $  \eta :   \mathop{\rm id} \rightarrow T $
 +
is a natural transformation. Further define $  \mu _ {A} : T T A \rightarrow T A $
 +
as the composite $  1 _ {TA} \circ 1 _ {TTA} $.  
 +
Then $  \mu $
 +
is also a natural transformation and $  ( T , \mu , \eta ) $
 +
is a triple. Moreover, this construction yielding a triple for each algebraic theory in clone form is a bijection. For a discussion of the algebraic theories (in clone and monad form) coming from a universal algebra and a third categorical way of viewing universal algebras see [[Universal algebra|Universal algebra]].
  
The language of categories and functors was originally introduced to meet the needs of algebraic topology and homological algebra [[#References|[a1]]], [[#References|[a4]]]. In the 1950's and early 1960's much attention was focused on Abelian categories (cf. [[Abelian category|Abelian category]]), which may be defined as categories satisfying all the elementary properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740409.png" />; it was shown in [[#References|[2]]] that they provide an adequate foundation for the development of homological algebra, and in [[#References|[a11]]] that every small Abelian category admits a full imbedding, preserving finite limits and colimits, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740410.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740411.png" />.
+
The language of categories and functors was originally introduced to meet the needs of algebraic topology and homological algebra [[#References|[a1]]], [[#References|[a4]]]. In the 1950's and early 1960's much attention was focused on Abelian categories (cf. [[Abelian category|Abelian category]]), which may be defined as categories satisfying all the elementary properties of $  \mathop{\rm Ab} $;  
 +
it was shown in [[#References|[2]]] that they provide an adequate foundation for the development of homological algebra, and in [[#References|[a11]]] that every small Abelian category admits a full imbedding, preserving finite limits and colimits, into $  \mathop{\rm Mod} _ {R} $
 +
for some $  R $.
  
In an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740412.png" />, the "Hom-sets" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740413.png" /> have a natural [[Abelian group|Abelian group]] structure; this observation provided one of the incentives for developing the theory of enriched (or relative) categories [[#References|[a12]]], that is, categories whose "Hom-sets" are objects of some "base category" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740414.png" />. Categories enriched over themselves (such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740415.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740416.png" />) are called closed categories [[#References|[a13]]] (cf. [[Closed category|Closed category]]); an important class of closed categories (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740417.png" /> but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740418.png" />) consists of those where the closed structure (the "internal Hom" ) is related by an adjunction to the categorical product structure — such categories are called Cartesian closed. The notion of a Cartesian closed category played an important role in F.W. Lawvere's axiomatization of the category of small categories as a foundation for mathematics [[#References|[a14]]], and in his latter development with M. Tierney of the notion of an elementary [[Topos|topos]], which has dominated much of categorical research in the 1970's and 1980's (see [[#References|[a15]]]). Cartesian closed categories are also of importance in logic, since they provide models for the (typed) [[Lambda-calculus|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740419.png" />-calculus]] (see [[#References|[a16]]]).
+
In an Abelian category $  \mathfrak C $,  
 +
the "Hom-sets" $  \mathfrak C ( A , B ) $
 +
have a natural [[Abelian group|Abelian group]] structure; this observation provided one of the incentives for developing the theory of enriched (or relative) categories [[#References|[a12]]], that is, categories whose "Hom-sets" are objects of some "base category" $  \mathfrak B $.  
 +
Categories enriched over themselves (such as $  \mathop{\rm Ab} $
 +
and $  \mathop{\rm Cat} $)  
 +
are called closed categories [[#References|[a13]]] (cf. [[Closed category|Closed category]]); an important class of closed categories (including $  \mathop{\rm Cat} $
 +
but not $  \mathop{\rm Ab} $)  
 +
consists of those where the closed structure (the "internal Hom" ) is related by an adjunction to the categorical product structure — such categories are called Cartesian closed. The notion of a Cartesian closed category played an important role in F.W. Lawvere's axiomatization of the category of small categories as a foundation for mathematics [[#References|[a14]]], and in his latter development with M. Tierney of the notion of an elementary [[Topos|topos]], which has dominated much of categorical research in the 1970's and 1980's (see [[#References|[a15]]]). Cartesian closed categories are also of importance in logic, since they provide models for the (typed) [[Lambda-calculus| $  \lambda $-
 +
calculus]] (see [[#References|[a16]]]).
  
Categories enriched over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740420.png" /> (commonly called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740422.png" />-categories) have also received a good deal of attention in recent years. They are distinguished from the general run of enriched categories by the possibility of considering diagrams within them which commute "up to isomorphism" but not exactly; the weaker notion of a [[Bicategory(2)|bicategory]] [[#References|[a17]]] is a further expression of this idea. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740423.png" />-categories and higher-dimensional categories have also been studied, and have proved to be of importance in the algebraic study of homotopy types [[#References|[a18]]]. In these areas of category theory coherence theorems play an important part: these are theorems which allow one to deduce the commutativity of a large class of diagrams from that of certain particular diagrams (see [[#References|[a19]]], for example).
+
Categories enriched over $  \mathop{\rm Cat} $(
 +
commonly called $  2 $-
 +
categories) have also received a good deal of attention in recent years. They are distinguished from the general run of enriched categories by the possibility of considering diagrams within them which commute "up to isomorphism" but not exactly; the weaker notion of a [[Bicategory(2)|bicategory]] [[#References|[a17]]] is a further expression of this idea. $  3 $-
 +
categories and higher-dimensional categories have also been studied, and have proved to be of importance in the algebraic study of homotopy types [[#References|[a18]]]. In these areas of category theory coherence theorems play an important part: these are theorems which allow one to deduce the commutativity of a large class of diagrams from that of certain particular diagrams (see [[#References|[a19]]], for example).
  
Further areas of category theory in which much work has been done in recent years include the theory of fibred categories [[#References|[a2]]] (which, together with enriched category theory, is an expression of the idea that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740424.png" /> can be replaced by some more general base category as a foundation for much of mathematics), and the theory of topological categories [[#References|[a3]]] (which is concerned with the study of concrete categories whose forgetful functors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740425.png" /> have good infinitary properties, similar to those of the forgetful functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740426.png" />, see also [[Topologized category|Topologized category]]).
+
Further areas of category theory in which much work has been done in recent years include the theory of fibred categories [[#References|[a2]]] (which, together with enriched category theory, is an expression of the idea that $  \mathop{\rm Set} $
 +
can be replaced by some more general base category as a foundation for much of mathematics), and the theory of topological categories [[#References|[a3]]] (which is concerned with the study of concrete categories whose forgetful functors to $  \mathop{\rm Set} $
 +
have good infinitary properties, similar to those of the forgetful functor $  \mathop{\rm Top} \rightarrow  \mathop{\rm Set} $,  
 +
see also [[Topologized category|Topologized category]]).
  
 
In addition to the books [[#References|[9]]] and [[#References|[11]]], [[#References|[12]]] and [[#References|[13]]] may also be recommended as general accounts of category theory.
 
In addition to the books [[#References|[9]]] and [[#References|[11]]], [[#References|[12]]] and [[#References|[13]]] may also be recommended as general accounts of category theory.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, "Natural isomorphisms in group theory" ''Proc. Nat. Sci. USA'' , '''28''' (1942) pp. 537–543 {{MR|0007421}} {{ZBL|0061.09203}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Bénabou, "Fibred categories and the foundations of naive category theory" ''J. Symbolic Logic'' , '''50''' (1985) pp. 10–37</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.C.L. Brümmer, "Topological categories" ''Topology Appl.'' , '''18''' (1984) pp. 27–41 {{MR|0759137}} {{ZBL|0551.18003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. MacLane, "Duality for groups" ''Bull. Amer. Math. Soc.'' , '''56''' (1950) pp. 485–516</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D.M. Kan, "Adjoint functors" ''Trans. Amer. Math. Soc.'' , '''87''' (1958) pp. 294–329 {{MR|0131451}} {{ZBL|0090.38906}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.G. Manes, "Algebraic theories" , Springer (1976) {{MR|0419557}} {{ZBL|0353.18007}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S. Eilenberg, J.C. Moore, "Adjoint functors and triples" ''Ill. J. Math.'' , '''9''' (1965) pp. 381–398 {{MR|0184984}} {{ZBL|0135.02103}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> F.E.J. Linton, "Some aspects of equational categories" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 84–94 {{MR|0209335}} {{ZBL|0201.35003}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F.W. Lawvere, "Functorial semantics of algebraic theories" ''Proc. Nat. Acad. Sci. USA'' , '''50''' (1963) pp. 869–872 {{MR|0158921}} {{ZBL|0119.25901}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> B. Pareigis, "Categories and functors" , Acad. Press (1970) {{MR|0265427}} {{MR|0265428}} {{ZBL|0211.32402}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> B. Mitchell, "The full imbedding theorem" ''Amer. J. Math.'' , '''86''' (1964) pp. 619–637 {{MR|0167511}} {{ZBL|0124.01502}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> G.M. Kelly, "Basic concepts of enriched category theory" , Cambridge Univ. Press (1982) {{MR|0651714}} {{ZBL|0478.18005}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> G.M. Kelly, "Closed categories" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 421–562 {{MR|0225841}} {{ZBL|0192.10604}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> F.W. Lawvere, "The category of categories as a foundation for mathematics" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 1–20 {{MR|0207517}} {{ZBL|0192.09702}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P.T. Johnstone, "Topos theory" , Acad. Press (1977) {{MR|0470019}} {{ZBL|0368.18001}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J. Lambek, P.J. Scott, "Introduction to higher-order categorical logic" , Cambridge Univ. Press (1986) {{MR|0856915}} {{ZBL|0596.03002}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J. Bénabou, "Introduction to bicategories" , ''Reports of the Midwest Category Seminar I'' , ''Lect. notes in math.'' , '''47''' , Springer (1967) pp. 1–77 {{MR|0220789}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> R. Brown, P.J. Higgins, "The equivalence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020740/c020740427.png" />-groupoids and crossed complexes" ''Cahiers Top. et Géom. Diff.'' , '''22''' (1981) pp. 371–386 {{MR|639048}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S. MacLane, "Why commutative diagrams coincide with equivalent proofs" , ''Algebraists' Homage'' , ''Contemp. Math.'' , '''13''' , Amer. Math. Soc. (1982) pp. 387–401 {{MR|}} {{ZBL|0504.18004}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Eilenberg, S. MacLane, "Natural isomorphisms in group theory" ''Proc. Nat. Sci. USA'' , '''28''' (1942) pp. 537–543 {{MR|0007421}} {{ZBL|0061.09203}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Bénabou, "Fibred categories and the foundations of naive category theory" ''J. Symbolic Logic'' , '''50''' (1985) pp. 10–37</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.C.L. Brümmer, "Topological categories" ''Topology Appl.'' , '''18''' (1984) pp. 27–41 {{MR|0759137}} {{ZBL|0551.18003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. MacLane, "Duality for groups" ''Bull. Amer. Math. Soc.'' , '''56''' (1950) pp. 485–516</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D.M. Kan, "Adjoint functors" ''Trans. Amer. Math. Soc.'' , '''87''' (1958) pp. 294–329 {{MR|0131451}} {{ZBL|0090.38906}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.G. Manes, "Algebraic theories" , Springer (1976) {{MR|0419557}} {{ZBL|0353.18007}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S. Eilenberg, J.C. Moore, "Adjoint functors and triples" ''Ill. J. Math.'' , '''9''' (1965) pp. 381–398 {{MR|0184984}} {{ZBL|0135.02103}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> F.E.J. Linton, "Some aspects of equational categories" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 84–94 {{MR|0209335}} {{ZBL|0201.35003}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F.W. Lawvere, "Functorial semantics of algebraic theories" ''Proc. Nat. Acad. Sci. USA'' , '''50''' (1963) pp. 869–872 {{MR|0158921}} {{ZBL|0119.25901}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> B. Pareigis, "Categories and functors" , Acad. Press (1970) {{MR|0265427}} {{MR|0265428}} {{ZBL|0211.32402}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> B. Mitchell, "The full imbedding theorem" ''Amer. J. Math.'' , '''86''' (1964) pp. 619–637 {{MR|0167511}} {{ZBL|0124.01502}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> G.M. Kelly, "Basic concepts of enriched category theory" , Cambridge Univ. Press (1982) {{MR|0651714}} {{ZBL|0478.18005}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> G.M. Kelly, "Closed categories" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 421–562 {{MR|0225841}} {{ZBL|0192.10604}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> F.W. Lawvere, "The category of categories as a foundation for mathematics" S. Eilenberg (ed.) et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer (1966) pp. 1–20 {{MR|0207517}} {{ZBL|0192.09702}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P.T. Johnstone, "Topos theory" , Acad. Press (1977) {{MR|0470019}} {{ZBL|0368.18001}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> J. Lambek, P.J. Scott, "Introduction to higher-order categorical logic" , Cambridge Univ. Press (1986) {{MR|0856915}} {{ZBL|0596.03002}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> J. Bénabou, "Introduction to bicategories" , ''Reports of the Midwest Category Seminar I'' , ''Lect. notes in math.'' , '''47''' , Springer (1967) pp. 1–77 {{MR|0220789}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a18]</TD> <TD valign="top"> R. Brown, P.J. Higgins, "The equivalence of $\infty$-groupoids and crossed complexes" ''Cahiers Top. et Géom. Diff.'' , '''22''' (1981) pp. 371–386 {{MR|639048}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S. MacLane, "Why commutative diagrams coincide with equivalent proofs" , ''Algebraists' Homage'' , ''Contemp. Math.'' , '''13''' , Amer. Math. Soc. (1982) pp. 387–401 {{MR|}} {{ZBL|0504.18004}} </TD></TR>
 +
</table>

Latest revision as of 09:17, 26 March 2023


A concept formalizing a number of algebraic properties of collections of morphism between mathematical objects of the same type (sets, topological spaces, groups, etc.) under the condition that these collections contain the identity mappings and are closed with respect to successive composition (or product) of mappings. A category $ \mathfrak C $ consists of a class $ \mathop{\rm Ob} \mathfrak C $, whose elements are called objects of the category, and a class $ \mathop{\rm Mor} \mathfrak C $, the elements of which are called morphisms of the category. These classes must satisfy the following conditions:

1) to each ordered pair of objects $ A , B $ is associated a set $ H _ {\mathfrak C } ( A , B ) $( also denoted by $ \mathfrak C ( A , B ) $, $ \mathop{\rm Hom} ( A , B ) $, or $ H ( A , B ) $) of members of $ \mathop{\rm Mor} $. If $ \alpha \in H ( A , B ) $, then $ A $ is called the source, or domain (of definition), of the morphism $ \alpha $, and $ B $ the target, or range (of values), or codomain of $ \alpha $; instead of $ \alpha \in H ( A , B ) $ one often writes $ \alpha : A \rightarrow B $ or $ A \rightarrow ^ \alpha B $;

2) each morphism of the category $ \mathfrak C $ belongs to only one set $ H _ {\mathfrak C } ( A , B ) $;

3) in the class $ \mathop{\rm Mor} \mathfrak C $, the following partial composition (product) rule is given: The product of two morphisms $ \alpha : A \rightarrow B $ and $ \beta : C \rightarrow D $ is defined if and only if $ B = C $, in which case it belongs to $ H ( A , D ) $; the product of $ \alpha $ and $ \beta $ is denoted by $ \alpha \beta $, or, depending on one's choice of conventions, by $ \beta \alpha $;

4) for any morphisms $ \alpha : A \rightarrow B $, $ \beta : B \rightarrow C $ and $ \gamma : C \rightarrow D $, the associativity law holds:

$$ ( \alpha \beta ) \gamma = \alpha ( \beta \gamma ) ; $$

5) each set $ H _ {\mathfrak C } ( A , A ) $ contains a (distinguished) morphism $ 1 _ {A} $ such that $ \alpha \cdot 1 _ {A} = \alpha $ and $ 1 _ {A} \cdot \beta = \beta $ for any morphisms $ \alpha : X \rightarrow A $ and $ \beta : A \rightarrow Y $; the morphisms $ 1 _ {A} $ are called units, identities or ones.

The notion of a class occurring in the definition of a category presupposes the use of axioms from set theory which distinguish between sets and classes. The most commonly used is the axiom scheme of Gödel–Bernays–von Neumann.

Sometimes, in the definition of a category, it is not required that the classes $ H ( A , B ) $ be sets. Sometimes, instead of using classes, one assumes the existence of a universal set and requires that the classes $ \mathop{\rm Ob} \mathfrak C $ and $ \mathop{\rm Mor} \mathfrak C $ belong to a fixed universal set.

Since there is a bijective correspondence between the identities of a category $ \mathfrak C $ and the class $ \mathop{\rm Ob} \mathfrak C $, a category $ \mathfrak C $ can be defined as a class of morphisms with a partial product satisfying additional conditions (see, for example, [6], [9]).

The notion of a category was introduced in 1945 [8]. The origins of category theory and the initial stimulus for its development came from algebraic topology. Subsequent investigation revealed the unifying role of the notion of a category and the notion of a functor related to it for many branches of mathematics.

Examples of categories.

1) The category of sets $ \mathop{\rm Ens} $. The class $ \textrm{ Ob Ens } $ consists of all sets; the class $ \textrm{ Mor Ens } $ consists of all functions between sets; composition of morphism is composition of functions (see Sets, category of).

2) The category of topological spaces $ \mathop{\rm Top} $( or $ \mathfrak T $). The class $ \textrm{ Ob Top } $ consists of all topological spaces, the class $ \textrm{ Mor Top } $ of all continuous mappings, while composition is again composition of mappings.

3) The category of groups $ \mathop{\rm Gr} $( or $ \mathfrak G $). The class $ \textrm{ Ob Gr } $ consists of all groups, the class $ \textrm{ Mor Gr } $ of all group homomorphisms, and composition again is composition of homomorphisms (see Category of groups). One defines in a similar fashion the categories of vector spaces over some field, the category of rings, etc.

4) The category of binary relations between sets $ \textrm{ Rel Ens } $( or $ R ( \mathfrak S ) $). The class of objects of this category is $ \textrm{ Ob Ens } $, but as morphisms from $ A $ to $ B $ one takes all binary relations, that is, all subsets of the Cartesian product $ A \times B $; composition is composition of binary relations (cf. Binary relation).

5) A monoid (semi-group with identity) is a category with a single object; conversely, every category consisting of a single object is a monoid.

6) A pre-ordered set $(N,{\le})$ can be regarded as a category $ \mathfrak N $ for which $ \mathop{\rm Ob} \mathfrak N = N $, $ \mathop{\rm Mor} \mathfrak N = \{ {( a , b ) } : {a , b \in N, a \leq b } \} $, while the product is defined by the equality $ ( a , b ) ( b , c ) = ( a , c ) $.

All categories listed above admit an "isomorphic imbedding" into the category of sets. A category with this property is called a concrete category. Not every category is concrete; for example, the category with as objects all topological spaces and whose morphisms are defined to be homotopy classes of continuous mappings [10].

The number of examples of categories can be considerably enlarged by means of various constructions; foremost, by means of functors of categories or categories of diagrams.

A mapping $ F : \mathfrak C \rightarrow \mathfrak C ^ { \prime } $ between two categories $ \mathfrak C $, $ \mathfrak C ^ { \prime } $ is called a covariant functor (cf. also Functor) if for each object $ A \in \mathop{\rm Ob} \mathfrak C $, $ F (A) \in \mathop{\rm Ob} \mathfrak C ^ { \prime } $, for each morphism $ \alpha \in H _ {\mathfrak C } ( A , B ) $, its image $ F ( \alpha ) \in H _ {\mathfrak C ^ { \prime } } ( F (A) , F (B) ) $, if $ F ( 1 _ {A} ) = 1 _ {F (A) } $, and if, finally, $ F ( \alpha \beta ) = F ( \alpha ) F ( \beta ) $ whenever the product $ \alpha \beta $ is defined. If the objects of $ \mathfrak C $ constitute a set, then one can construct the category of diagrams $ \mathop{\rm Funct} ( \mathfrak C , \mathfrak C ^ { \prime } ) $ or $ F ( \mathfrak C , \mathfrak C ^ { \prime } ) $, the objects of which are all covariant functors from $ \mathfrak C $ to $ \mathfrak C ^ { \prime } $ and with as morphisms all natural transformations of these functors.

To each category $ \mathfrak C $ one can associate its dual category $ \mathfrak C $, or $ \mathfrak C ^ {T} $, or $ \mathfrak C ^ {op} $, for which $ \mathop{\rm Ob} \mathfrak C ^ {*} = \mathop{\rm Ob} \mathfrak C $ and $ H _ {\mathfrak C ^ {*} } ( A , B ) = H _ {\mathfrak C } ( B , A ) $ for any $ A , B \in \mathop{\rm Ob} \mathfrak C $. A covariant functor from $ \mathfrak C ^ {*} $ to $ \mathfrak C ^ { \prime } $ is called a contravariant functor from $ \mathfrak C $ to $ \mathfrak C ^ { \prime } $. Along with functors of one argument one can consider many-placed functors or functors of several arguments; cf. Functor.

For each statement in category theory there is a dual statement, obtained by a formal "reversal of the arrows" . In this connection the so-called duality principle holds: A statement $ p $ is true in category theory if and only if the dual statement $ p ^ {*} $ is true.

Many concepts and results in mathematics turn out to be dual to others from a category-theoretic point of view: injectivity and projectivity, nilpotency and the notion of category of a topological space in the sense of Lyusternik–Shnirel'man, varieties and radicals in algebra, etc.

A category-theoretic analysis of the foundations of homology theory led to the introduction in the mid-fifties of so-called Abelian categories (cf. Abelian category). Within this framework it proved possible to realize the basic constructions of homological algebra [2]. In the 1960s, interest in non-Abelian categories grew, as a result of problems in logic, general algebra, topology, and algebraic geometry. An intensive development of universal algebra and the axiomatic construction of homotopy theory marked the beginnings of various lines of investigation: the category-theoretic study of varieties of universal algebras, the theory of isomorphisms of direct decompositions, the theory of adjoint functors, and the theory of duality of functors. The latter development uncovered the existence of relations between these areas of study. As a result of the recent theory of relative categories, which makes wide use of the techniques of adjoint functors and closed categories, a duality has been established between homotopy theory and the theory of universal algebras; this is based on the interpretation of the categoric definitions of a monoid and a comonoid in suitable categories of functors (see, for example, [7]). Along with the development of the general theory of relative categories special classes of such categories were introduced: $ 2 $- categories or formal categories; categories with an involution or $ 1 $- categories, including, in particular, the category of binary relations; etc. A special case of $ 2 $- categories is the category of small categories, which can be placed at the basis of an axiomatic construction of mathematics.

The classes of categories listed above are characterized by the fact that their sets of morphisms $ H ( A , B ) $ possess an additional structure. Another method of introducing an additional structure in a category is to provide the category with a Grothendieck topology and to construct the category of sheaves over the topologized category (so-called topoi; cf. Topos).

References

[1] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) MR0236236 Zbl 0197.29205
[2] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. , 9 (1957) pp. 119–221 MR0102537
[3] A.G. Kurosh, A.Kh. Livshits, E.G. Shul'geifer, "Foundations of category theory" Uspekhi Mat. Nauk , 15 : 6 (1960) pp. 3–52 (In Russian)
[4] Itogi Nauk. Algebra. Topol. 1962 (1963) pp. 90–106
[5] Itogi Nauk. Algebra. Topol. Geom. 1967 (1969) pp. 9–57
[6] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)
[7] M.C. Bunge, "Relative functor categories and categories of algebras" J. of Algebra , 11 (1969) pp. 64–101 MR0236238 Zbl 0165.32902
[8] S. Eilenberg, S. MacLane, Trans. Amer. Math. Soc. , 58 (1945) pp. 231–294
[9] P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper & Row (1964) MR0166240 Zbl 0121.02103
[10] P. Freyd, "On the concreteness of certain categories" , Symp. Math. , 4 , Acad. Press (1970) pp. 431–456 MR0271184 Zbl 0248.18008
[11] S. MacLane, "Categories for the working mathematician" , Springer (1971) Zbl 0232.18001
[12] H. Schubert, "Categories" , 1–2 , Springer (1972) MR0349793 Zbl 0253.18002
[13] B. Mitchell, "Theory of categories" , Acad. Press (1965) MR0202787 Zbl 0136.00604

Comments

The notion of a category was introduced in 1942 by S. Eilenberg and S. MacLane [a1], and has since found numerous applications in algebra, topology and the foundations of mathematics. The intuitive idea is that a category consists of all the objects in some "universe of mathematical discourse" together with all the mappings between them. By identifying an object with its identity morphism it is possible to define the notion of a category in terms of morphisms alone. It is intuitively clearer and also more customary to use both objects and morphisms. The two approaches are mixed-up to some extent in the article above. The object and morphism definition is as follows. A category $ \mathfrak C $ consists of

A1) a class $ \mathop{\rm Ob} \mathfrak C $ whose elements are called objects of $ \mathfrak C $, and a class $ \mathop{\rm Mor} \mathfrak C $ whose elements are called morphisms or arrows of $ \mathfrak C $;

A2) operations assigning to each morphism $ \alpha $ of $ \mathfrak C $ a pair of objects $ ( d _ {0} ( \alpha ) , d _ {1} ( \alpha ) ) $, called the domain and codomain (or source and target) of $ \alpha $. One writes "a: A B" or "AaB" to mean "a is a morphism with domain A and codomain B" ; this rephrases 1) above;

A3) an operation assigning to each object $ A $ of $ \mathfrak C $ a morphism $ 1 _ {A} : A \rightarrow A $, called the identity morphism on $ A $; this is the precise meaning of part of 5) above;

A4) a partial (binary) product operation for morphisms, the product $ \alpha \beta $( called the composite of $ \alpha $ and $ \beta $) being defined if and only if $ d _ {0} ( \alpha ) = d _ {1} ( \beta ) $, and satisfying $ d _ {1} ( \alpha \beta ) = d _ {1} ( \alpha ) $ and $ d _ {0} ( \alpha \beta ) = d _ {0} ( \beta ) $ whenever it is defined; this rephrases 3);

these data being subject to the axioms

A5) composition is associative, i.e. $ ( \alpha \beta ) \gamma = \alpha ( \beta \gamma ) $ whenever both sides are defined; this rephrases 4) above;

A6) identity morphisms are units for composition, i.e. $ \alpha 1 _ {A} = \alpha $ and $ 1 _ {A} \beta = \beta $ whenever the composites are defined; this and A3) rephrase 5).

The classes $ \mathop{\rm Ob} \mathfrak C $ and $ \mathop{\rm Mor} \mathfrak C $ are not required to be sets, and in many of the leading examples (see, e.g. the main text above and the examples (A1)–(A4), (A7) below) they are not sets. However, most examples have the property that, for each pair of objects $ ( A , B ) $, the collection of morphisms $ \alpha $ with $ d _ {0} ( \alpha ) = A $ and $ d _ {1} ( \alpha ) = B $ forms a set (usually denoted by $ \mathfrak C ( A , B ) $ or $ \mathop{\rm Hom} ( A , B ) $); such categories are sometimes called locally small, although other writers include this condition as part of the definition of a category. A category $ \mathfrak C $ is said to be small if $ \mathop{\rm Ob} \mathfrak C $ and $ \mathop{\rm Mor} \mathfrak C $ are sets (cf. Small category); it turns out that many of the fundamental mathematical structures may be regarded as small categories (e.g. (A6) and (A7) below).

It follows from the definition that each object in a category has a unique identity morphism; thus it is possible to identify objects with their identity morphisms, leading to an axiomatization of categories in which "morphism" and "composite" are the only primitive notions (see [9]).

Examples of categories.

(A1) See the main article. The category $ \mathop{\rm Ens} $ of sets is more often denoted by $ \mathop{\rm Set} $.

(A3) Next to the category $ \mathop{\rm Gr} $ one may consider the categories $ \mathop{\rm Ab} $ of Abelian groups, $ \mathop{\rm Mod} _ {R} $ of (right) modules over a fixed ring $ R $, etc.

(A4) The category of binary relations between sets is usually denoted by $ \mathop{\rm Rel} $.

(A5) A semi-group with identity $ M $ is also called a monoid. It defines a category with one object $ * $, the elements of $ M $ being interpreted as morphisms $ * \rightarrow * $.

(A6) A partially ordered set $ ( A , \leq ) $ defines a category whose objects are the elements of $ A $, and whose morphisms are the instances of the order-relation: that is, there is just one morphism $ a \rightarrow b $ if $ a \leq b $, and none otherwise.

(A7) Similarly, an oriented graph (or diagram scheme) can be interpreted as a category of which the objects are the vertices and the morphism from $ v _ {1} $ to $ v _ {2} $ are the oriented paths from $ v _ {1} $ to $ v _ {2} $ including the trivial identity paths from $ v $ to $ v $ for all vertices. Inversely a category can be seen as a (very large) oriented graph with loops and multiple edges together with an equivalence relation identifying certain paths, cf. [11], Section II.7.

(A8) The homotopy category $ \mathop{\rm Htpy} $ or $ \mathop{\rm Htp} $ has the same objects as $ \mathop{\rm Top} $, but its morphisms are homotopy classes of continuous mappings. This category can be proved to be nonconcrete.

In the study of categories, functors (morphisms of categories) play an essential role. A functor $ F : \mathfrak C \rightarrow \mathfrak D $ consists of two functions, one assigning to each object $ A $ of $ \mathfrak C $ an object $ FA $ of $ \mathfrak D $, and the other assigning to each morphism $ \alpha $ of $ \mathfrak C $ a morphism $ F \alpha $ of $ \mathfrak D $, in such a way that the categorical structure is preserved: $ d _ {i} ( F \alpha ) = F ( d _ {i} ( \alpha ) ) $ $ ( i = 0 , 1 ) $, $ F ( 1 _ {A} ) = 1 _ {FA} $ and $ F ( \alpha \beta ) = ( F \alpha ) ( F \beta ) $ whenever $ \alpha \beta $ is defined. There is also a third level of structure: if $ F $ and $ G $ are both functors $ \mathfrak C \rightarrow \mathfrak D $, a natural transformation (or functorial morphism $ \eta : F \rightarrow G $ is a function assigning to each object $ A $ of $ \mathfrak C $ a morphism $ \eta _ {A} : F A \rightarrow G A $ in $ \mathfrak D $, such that for every $ A \rightarrow ^ \alpha B $ in $ \mathfrak C $ the diagram

$$ \begin{array}{rcl} FA & \mathop \rightarrow \limits ^ { {F \alpha }} &FB \\ \eta _ {A} \downarrow &{} &\downarrow \eta _ {B} \\ GA & \mathop \rightarrow \limits _ { {G \alpha }} &GB \\ \end{array} $$

commutes (i.e. $ ( G \alpha ) ( \eta _ {A} ) = ( \eta _ {B} ) ( F \alpha ) $). Functors may be composed (and every category has an identity functor); thus there is a category $ Cat $ of (small) categories and functors between them. Natural transformations may be composed; thus, given two categories $ \mathfrak C $ and $ \mathfrak D $, there is a category $ [ \mathfrak C , \mathfrak D ] $( or $ \mathfrak D ^ {\mathfrak C } $) of functors $ \mathfrak C \rightarrow \mathfrak D $ and natural transformations between them. This is one important way in which new categories are constructed from existing ones. The resulting categories are called categories of functors or categories of diagrams. The latter name is especially understandable if $ \mathfrak C $ is the category corresponding to a diagram scheme (oriented graph). Indeed, then a functor $ \mathfrak C \rightarrow \mathfrak D $" is" a diagram in $ \mathfrak D $. Other important constructions to obtain new categories are: taking quotients (cf. Quotient category), taking localizations (cf. Localization in categories) and constructing relative and comma categories. The important notion of a derived category involves several of these constructions. If $ \mathfrak C $ is a category and $ B $ an object of $ \mathfrak C $, then the relative category $ \mathfrak C / B $ of objects over $ B $ has as objects all morphisms $ \alpha : A \rightarrow B $ of $ \mathfrak C $ into $ B $ and a morphism in $ \mathfrak C / \mathfrak B $ of $ \alpha $ to $ \alpha ^ \prime : A ^ { \prime } \rightarrow B $ is a morphism $ \phi : A \rightarrow A ^ { \prime } $ such that $ \alpha ^ \prime \circ \phi = \alpha $. Dually there is the notion of the relative category of objects in $ \mathfrak C $ under a given object. Intuitively an object $ \alpha : A \rightarrow B $ of $ \mathfrak C / B $ is a family of objects of $ \mathfrak C $ parametrized by $ B $ or a fibre object (fibred object). The systematic consideration of these relative objects, i.e. fibre objects (and their duals) combined with base change and deformation ideas has become a most important technique in many parts of mathematics, especially in algebra (notably homological algebra), algebraic and differential geometry, topology, and differential and algebraic topology. It is especially important to find the right fibrewise versions of definitions, theorems and concepts. (A second additional not unrelated major trend involves finding the right equivariant versions in those case in which there is a group of symmetries present (as well).)

The idea of a comma category generalizes that of categories of objects over or under a given object. Let there be given three categories $ \mathfrak C $, $ \mathfrak B $, $ \mathfrak A $ and two functors $ S , T $ arranged as follows

$$ \mathfrak A \rightarrow ^ { T } \mathfrak C \leftarrow ^ { S } \mathfrak B . $$

Then the comma category $ ( T , S ) $ or $ ( T \downarrow S ) $ has as objects all triples $ ( a , f , b ) $ consisting of an object $ a \in \mathfrak A $, an object $ b \in \mathfrak B $ and a morphism $ f : T A \rightarrow S B $ in $ \mathfrak C $. A morphism $ ( \alpha , \beta ) $ from $ ( a , f , b ) $ to $ ( a ^ \prime , f ^ { \prime } , b ^ \prime ) $ consists of a pair of morphisms $ \alpha : A \rightarrow A ^ { \prime } $ and $ \beta : B \rightarrow B ^ { \prime } $ such that $ f ^ { \prime } \circ T \alpha = T \beta \circ f $.

Examples of functors. ITEM {(A9} The forgetful functor $ \mathop{\rm Top} \rightarrow \mathop{\rm Set} $ sends each topological space to its underlying set, and each continuous function to itself ( "forgetting" the continuity). Similarly, one has forgetful functors $ \mathop{\rm Gr} \rightarrow \mathop{\rm Set} $, $ \mathop{\rm Mod} _ {R} \rightarrow \mathop{\rm Ab} $, etc.

(A10) If $ \mathfrak C $ is a locally small category, then for each $ A \in \mathop{\rm Ob} \mathfrak C $ there is a functor $ \mathfrak C ( A , - ) : \mathfrak C \rightarrow \mathop{\rm Set} $ sending $ B $ to the set $ \mathfrak C ( A , B ) $ and $ B \rightarrow ^ \alpha B ^ { \prime } $ to the function which sends $ \beta $ to $ \alpha \beta $. Such functors (or functors isomorphic to them in $ [ \mathfrak C , \mathop{\rm Set} ] $) are called representable (cf. Representable functor).

(A11) There is a functor $ F : \mathop{\rm Set} \rightarrow \mathop{\rm Gr} $ sending a set $ A $ to the free group generated by $ A $, and a function $ A \rightarrow ^ \alpha B $ to the unique homomorphism $ F A \rightarrow F B $ sending the generator $ a $ of $ F A $ to $ \alpha (a) \in F B $, for each $ a \in A $.

(A12) The (singular) homology groups of spaces (cf. Homology group) define functors $ \mathop{\rm Htpy} \rightarrow \mathop{\rm AB} $( one for each dimension $ \geq 0 $).

(A13) A functor between monoids, considered as categories, is just a monoid homomorphism.

(A14) A functor between partially ordered sets, considered as categories, is just an order-preserving mapping.

(A15) If $ G $ is a group, considered as a category, a functor $ G \rightarrow \mathop{\rm Set} $( respectively, $ G \rightarrow \mathop{\rm Mod} _ {R} $) is a permutation (respectively, an $ R $- linear representation) of $ G $( cf. Representation of a group).

A functor $ F : \mathfrak C \rightarrow \mathfrak D $ is said to be faithful (cf. Faithful functor) if it is "injective on Hom-sets" ; i.e. if, given two morphisms $ A \rightarrow ^ \alpha B $ and $ A \rightarrow ^ \beta B $ with the same domain and codomain in $ \mathfrak C $, $ F \alpha = F \beta $ implies $ \alpha = \beta $. $ F $ is said to be full if it is "surjective on Hom-sets" in a similar sense. Forgetful functors (as in (A9) above) are always faithful. The property that a category $ \mathfrak C $ be concrete can now be rephrased as: There is a faithful functor $ \mathfrak C \rightarrow \mathop{\rm Set} $.

Given a category $ \mathfrak C $, one can form its opposite or dual category $ \mathfrak C ^ {op} $ by keeping the same objects as $ \mathfrak C $ and reversing all the morphisms. The category $ \mathop{\rm Rel} $ is isomorphic to its opposite, though most familiar categories are not. A functor $ \mathfrak C ^ {op} \rightarrow \mathfrak D $ is sometimes called a contravariant functor from $ \mathfrak C $ to $ \mathfrak D $; for emphasis, functors $ \mathfrak C \rightarrow \mathfrak D $ are then called covariant. (For example, if $ \mathfrak C $ is locally small, one may define a contravariant functor $ \mathfrak C ( - , A ) $ from $ \mathfrak C $ to $ \mathop{\rm Set} $, by analogy with the covariant functor $ \mathfrak C ( A , - ) $ of example (A10) above.) The duality principle for categories is essentially the assertion that something which is true for all categories is true for the duals of all categories.

S. MacLane [a4] introduced the idea that Cartesian products can be characterized in categorical terms, by a universal property; this gave rise to the general categorical notion of limit (and the dual notion of colimit), which includes products as a special case (cf. Limit and Universal problems).

The key notion of adjunction came latter [a5]: given functors $ F : \mathfrak C \rightarrow \mathfrak D $ and $ G : \mathfrak D \rightarrow \mathfrak C $, one says that $ F $ is left adjoint to $ G $( written $ F \dashv G $) if there is a bijection between morphisms $ F A \rightarrow B $ and morphisms $ A \rightarrow G B $ which is natural in $ A $ and $ B $; this is equivalent to the existence of natural transformations $ \eta : 1 _ {\mathfrak C } \rightarrow G F $ and $ \epsilon : F G \rightarrow 1 _ {\mathfrak D } $ satisfying certain identities [11] (cf. Adjoint functor). For example, the free group functor (example (A11) above) is left adjoint to the forgetful functor $ \mathop{\rm Gr} \rightarrow \mathop{\rm Set} $; Galois connections (cf. Galois correspondence) are examples of (contravariant) adjunctions between partially ordered sets. A functor which has a left adjoint preserves all limits; the converse implication is valid under suitable "smallness conditions" (the adjoint functor theorem, see [9]).

Given an adjunction $ ( F \dashv G ) $ as above, the composite functor $ T = G F : \mathfrak C \rightarrow \mathfrak C $ is equipped with natural transformations $ \eta : 1 _ {\mathfrak C } \rightarrow T $ and $ \mu = G \epsilon _ {F} : T T \rightarrow T $ satisfying certain identities; these data define the notion of a monad or triple on a category, which played a central role in much categorical research in the 1960's and later years.

The identities which a triple $ ( T , \mu , \eta ) $ on a category $ \mathfrak C $ is required to satisfy are the following: $ \mu _ {A} \circ T ( \mu _ {A} ) = \mu _ {A} \circ \mu _ {T(A)} $, $ \mu _ {A} \circ T ( \eta _ {A} ) = 1 _ {T (A) } $, $ \mu _ {A} \circ \eta _ {T (A) } = 1 _ {T (A) } $. An algebra for the triple $ T $, or $ T $- algebra, is an object $ X $ of $ \mathfrak C $ together with a morphism $ \xi : T X \rightarrow X $ such that the following identities hold: $ \xi \circ \eta _ {X} = 1 _ {X} $, $ \xi \circ T ( \xi ) = \xi \circ \mu _ {X} $. It is a good idea to write out these requirements in terms of commutative diagrams. They are reminiscent of associativity and unit requirements.

Dually, i.e. reversing all arrows, there is the notion of a cotriple and the corresponding notion of a co-algebra over such a cotriple. An important example of a cotriple in the category $ \mathop{\rm Ring} $ of commutative rings with unit element is the functor $ W : \mathop{\rm Ring} \rightarrow \mathop{\rm Ring} $ of the big Witt vectors together with the structure of a special $ \lambda $- ring on $ W (R) $. The co-algebras for this cotriple are precisely the special $ \lambda $- algebras (cf. Witt vector and $ \lambda $- ring).

Important examples of triples arise from adjunctions involving forgetful functors. For example, let $ V : \mathop{\rm Ring} \rightarrow \mathop{\rm Set} $ be the forgetful functor from the category of commutative rings with unit element to the category of sets. This one has an adjoint $ F : \mathop{\rm Set} \rightarrow \mathop{\rm Ring} $ which assigns to a set $ E \in \mathop{\rm Set} $ the free commutative ring with generator $ E $, i.e. the ring $ \mathbf Z [ X _ {e} : e \in E ] $ of commutative polynomials over $ \mathbf Z $ in the variables $ X _ {e} $, $ e \in E $. The freeness property of $ \mathbf Z [ X _ {e} : e \in E ] $, i.e. the property that for every ring $ A $ and every collection of elements $ ( a _ {e} ) _ {e \in E } $ of $ A $ there is precisely one homomorphism of rings $ \phi : \mathbf Z [ X _ {e} : e \in E ] \rightarrow A $ such that $ \phi ( X _ {e} ) = a _ {e} $ for all $ e \in E $, precisely expresses the fact that $ V $ and $ F $ are adjoint functors: $ \mathop{\rm Set} ( E , V (A) ) \cong \mathop{\rm Ring} ( F E , A ) $. The corresponding natural transformation $ \mathop{\rm id} \rightarrow V F $ is given by $ e \mapsto X _ {e} $( cf. also Adjoint functor).

Every monad and comonad can be induced by an adjunction; in fact there is a "best possible" such adjunction, in which $ \mathfrak D $ is taken to be the category of (Eilenberg–Moore) algebras for the monad, [a7]. A general adjunction $ (F \dashv G ) $ is said to be monadic (or $ \mathfrak D $ is said to be monadic over $ \mathfrak C $) if $ \mathfrak D $ is (canonically) equivalent to the category of algebras for the induced monad on $ \mathfrak C $. The adjunction between the $ \mathop{\rm Set} $ and $ \mathop{\rm Gr} $, mentioned above, is monadic; more generally, the categories which are monadic over $ \mathop{\rm Set} $ can be characterized [a8] as those which arise from varieties of universal algebras (provided one allows infinitary as well as finitary algebraic operations; the finitary case can also be characterized in categorical terms [9], using the notion of algebraic theory). See also Variety of universal algebras.

Another phrase that is used to denote a triple $ ( T , \mu , \eta ) $ is algebraic theory (in monad form) over the category $ \mathfrak C $. It is so to speak the theory of the category of $ T $- algebras. There are, at least, two more equivalent ways in which this notion is approached. One is as follows [a6]. An algebraic theory in clone form $ ( T , \eta , \circ ) $ consists of an "object assignment function" $ A \mapsto T A $( $ = $ $ T $- terms with variables in $ A $) for all objects $ A $, an "insertion of variables mapping" $ \eta _ {A} : A \rightarrow T A $ for all $ A $ and a "clone-composition function" $ \mathfrak C ( B , T C ) \times \mathfrak C ( A , T B ) \rightarrow ^ \circ \mathfrak C ( A , T C ) $ for each ordered triple $ ( A , B , C ) $ of objects of $ \mathfrak C $. For each $ f : A \rightarrow B $ in $ \mathfrak C $ let $ f ^ { \# } $ be the composite $ A \rightarrow B \rightarrow ^ {\eta _ {B} } T B $. Then the data $ ( T , \eta , \circ ) $ are supposed to satisfy the following axioms. For all $ \alpha : A \rightarrow T B $, $ \beta : B \rightarrow T C $, $ \gamma : C \rightarrow T D $ and $ f : A \rightarrow B $,

$$ ( \gamma \circ \beta ) \circ \alpha = \gamma \circ ( \beta \circ \alpha ) , $$

$$ \eta _ {B} \circ \alpha = \alpha , $$

$$ \alpha \circ f ^ { \# } = \alpha f . $$

This defines a new category $ \mathfrak C _ {T} $, the Kleishi category of $ ( T , \eta , \circ ) $. The objects of $ \mathfrak C _ {T} $ are the objects of $ \mathfrak C $, $ \mathfrak C _ {T} ( A , B ) = \mathfrak C ( A , T B ) $, composition is given by $ \circ $, and the identity morphisms are the $ \eta _ {A} $ in $ \mathfrak C _ {T} ( A , A ) = \mathfrak C ( A , T A ) $.

A simple example of an algebraic theory in clone form is as follows. Let $ R $ be a ring with unit. For a set $ A $ let $ T A $ be the vector space $ R ^ {(A)} = \{ {( r _ {a} ) _ {a \in A } } : {r _ {a} \in R \textrm{ and only finitely many } r _ {a } \textrm{ are different "_" zero } } \} $. A matrix with columns indexed by $ B $ and rows indexed by $ A $ is a mapping $ \alpha : B \rightarrow T A $, i.e. a morphism in $ \mathop{\rm Set} $; $ \alpha (b) $ is the $ b $- th column of the matrix. Given an $ A \times B $ matrix $ \alpha : B \rightarrow T A $ and a $ B \times C $ matrix $ \beta : C \rightarrow T B $, define their composite $ \alpha \circ \beta : C \rightarrow T A $ by the usual matrix product, i.e. $ ( \alpha \circ \beta ) (c) $ is the $ A $- vector with components

$$ ( \alpha \circ \beta ) (c) _ {a} = \ \sum _ { b } \alpha (b) _ {a} \beta (c) _ {b} . $$

The insertion of variables assignment $ \eta : A \rightarrow T A $ is defined by $ \eta (a) _ {a ^ \prime } = \delta _ {a , a ^ \prime } $ where $ \delta $ denotes the Kronecker delta (cf. Kronecker symbol). It is easily checked that the axioms above are satisfied.

Let $ ( T , \eta , \circ ) $ be an algebraic theory in clone form. For $ f : A \rightarrow B $ in $ \mathfrak C $ define $ T f : T A \rightarrow T B $ as the composite $ f ^ { \# } \circ \mathop{\rm id} _ {TA} $. It follows readily that $ T $ is then a functor and that $ \eta : \mathop{\rm id} \rightarrow T $ is a natural transformation. Further define $ \mu _ {A} : T T A \rightarrow T A $ as the composite $ 1 _ {TA} \circ 1 _ {TTA} $. Then $ \mu $ is also a natural transformation and $ ( T , \mu , \eta ) $ is a triple. Moreover, this construction yielding a triple for each algebraic theory in clone form is a bijection. For a discussion of the algebraic theories (in clone and monad form) coming from a universal algebra and a third categorical way of viewing universal algebras see Universal algebra.

The language of categories and functors was originally introduced to meet the needs of algebraic topology and homological algebra [a1], [a4]. In the 1950's and early 1960's much attention was focused on Abelian categories (cf. Abelian category), which may be defined as categories satisfying all the elementary properties of $ \mathop{\rm Ab} $; it was shown in [2] that they provide an adequate foundation for the development of homological algebra, and in [a11] that every small Abelian category admits a full imbedding, preserving finite limits and colimits, into $ \mathop{\rm Mod} _ {R} $ for some $ R $.

In an Abelian category $ \mathfrak C $, the "Hom-sets" $ \mathfrak C ( A , B ) $ have a natural Abelian group structure; this observation provided one of the incentives for developing the theory of enriched (or relative) categories [a12], that is, categories whose "Hom-sets" are objects of some "base category" $ \mathfrak B $. Categories enriched over themselves (such as $ \mathop{\rm Ab} $ and $ \mathop{\rm Cat} $) are called closed categories [a13] (cf. Closed category); an important class of closed categories (including $ \mathop{\rm Cat} $ but not $ \mathop{\rm Ab} $) consists of those where the closed structure (the "internal Hom" ) is related by an adjunction to the categorical product structure — such categories are called Cartesian closed. The notion of a Cartesian closed category played an important role in F.W. Lawvere's axiomatization of the category of small categories as a foundation for mathematics [a14], and in his latter development with M. Tierney of the notion of an elementary topos, which has dominated much of categorical research in the 1970's and 1980's (see [a15]). Cartesian closed categories are also of importance in logic, since they provide models for the (typed) $ \lambda $- calculus (see [a16]).

Categories enriched over $ \mathop{\rm Cat} $( commonly called $ 2 $- categories) have also received a good deal of attention in recent years. They are distinguished from the general run of enriched categories by the possibility of considering diagrams within them which commute "up to isomorphism" but not exactly; the weaker notion of a bicategory [a17] is a further expression of this idea. $ 3 $- categories and higher-dimensional categories have also been studied, and have proved to be of importance in the algebraic study of homotopy types [a18]. In these areas of category theory coherence theorems play an important part: these are theorems which allow one to deduce the commutativity of a large class of diagrams from that of certain particular diagrams (see [a19], for example).

Further areas of category theory in which much work has been done in recent years include the theory of fibred categories [a2] (which, together with enriched category theory, is an expression of the idea that $ \mathop{\rm Set} $ can be replaced by some more general base category as a foundation for much of mathematics), and the theory of topological categories [a3] (which is concerned with the study of concrete categories whose forgetful functors to $ \mathop{\rm Set} $ have good infinitary properties, similar to those of the forgetful functor $ \mathop{\rm Top} \rightarrow \mathop{\rm Set} $, see also Topologized category).

In addition to the books [9] and [11], [12] and [13] may also be recommended as general accounts of category theory.

References

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How to Cite This Entry:
Category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Category&oldid=38856
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article