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''in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315601.png" />''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315602.png" /> of an oriented graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315603.png" /> with set of vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315604.png" /> and set of edges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315605.png" /> into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315606.png" /> for which
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/d/d031/d031560/d0315607.png" /></td> </tr></table>
+
''in a category  $  C $''
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315608.png" /> if the edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d0315609.png" /> has source (origin) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156010.png" /> and target (end) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156011.png" />. The concept of a diagram in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156012.png" /> may also be defined as the image of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156013.png" />, in order to obtain a better visualization of diagrams.
+
A mapping  $  D $
 +
of an oriented graph  $  \Gamma $
 +
with set of vertices  $  I $
 +
and set of edges  $  U $
 +
into the category  $  C $
 +
for which
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156014.png" /> be an oriented chain of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156015.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156016.png" /> and target <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156017.png" />, i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156018.png" /> denote the composition of morphisms
+
$$
 +
D ( I)  \subset    \mathop{\rm Ob} ( C) ,\  D ( U)  \subset    \mathop{\rm Mor} ( C) ,
 +
$$
  
<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/d/d031/d031560/d03156019.png" /></td> </tr></table>
+
and  $  D ( u) \in  \mathop{\rm Hom} ( D ( i) , D ( j) ) $
 +
if the edge  $  u \in U $
 +
has source (origin)  $  i $
 +
and target (end)  $  j $.  
 +
The concept of a diagram in  $  C $
 +
may also be defined as the image of the mapping  $  D $,
 +
in order to obtain a better visualization of diagrams.
  
The diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156020.png" /> is said to be commutative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156021.png" /> for any two oriented chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156023.png" /> with identical source and target.
+
Let  $  \phi = ( u _ {1} \dots u _ {n} ) $
 +
be an oriented chain of the graph  $  \Gamma $
 +
with source  $  i $
 +
and target  $  j $,
 +
i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let  $  D ( \phi ) :  D ( i) \rightarrow D ( j) $
 +
denote the composition of morphisms
 +
 
 +
$$
 +
D ( u _ {n} )  \circ \dots \circ  D ( u _ {1} ) .
 +
$$
 +
 
 +
The diagram  $  D $
 +
is said to be commutative if $  D ( \phi ) = D ( \phi  ^  \prime  ) $
 +
for any two oriented chains $  \phi $
 +
and $  \phi  ^  \prime  $
 +
with identical source and target.
  
 
The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form
 
The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form
  
<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/d/d031/d031560/d03156024.png" /></td> </tr></table>
+
$$
 +
\cdot _ { i _ {1} } \rightarrow ^ { {u _ 1 } } \dots \rightarrow \cdot _ {i _ {n-} 1 }
 +
\rightarrow ^ { {u _ n-} 1 } \cdot _ {i _ {n} } .
 +
$$
  
 
The corresponding diagram is represented as follows:
 
The corresponding diagram is represented 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/d/d031/d031560/d03156025.png" /></td> </tr></table>
+
$$
 +
A _ {1}  \rightarrow ^ { {f _ 1} }  A _ {2}  \rightarrow \dots \rightarrow  A _ {n-} 1  \mathop \rightarrow \limits ^ { {f _ {n-} 1 }}  A _ {n} ,
 +
$$
 +
 
 +
where the  $  A _ {k} = D ( i _ {k} ) $
 +
are objects in the category  $  C $,
 +
while the  $  f _ {k} = D ( u _ {k} ) $
 +
are morphisms in this category.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156026.png" /> are objects in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156027.png" />, while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156028.png" /> are morphisms in this category.
+
A triangular diagram in a category $  C $
 +
corresponds to the graph
  
A triangular diagram in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156029.png" /> corresponds to the graph
+
$$
  
<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/d/d031/d031560/d03156030.png" /></td> </tr></table>
+
\begin{array}{lcr}
 +
{}  &i _ {1}  & \mathop \rightarrow \limits _ { {u _ {3} }}  \\
 +
{} _ {u _ {1}  }  \nearrow  &i _ {2}  &\searrow _ {u _ {2}  }  \\
 +
i _ {1}  &{}  \\
 +
\end{array}
 +
 
 +
$$
  
 
and is represented as follows:
 
and is represented 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/d/d031/d031560/d03156031.png" /></td> </tr></table>
+
$$
  
Commutativity of this diagram means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156032.png" />.
+
\begin{array}{lcr}
 +
A _ {1}  &\rightarrow ^ { {f _ 1} }  &A _ {2}  \\
 +
{} _ {f _ {3}  }  \searrow  &{}  &\swarrow _ {f _ {2}  }  \\
 +
{}  &A _ {3}  &{}  \\
 +
\end{array}
 +
 
 +
$$
 +
 
 +
Commutativity of this diagram means that $  f _ {3} = f _ {2} \circ f _ {1} $.
  
 
A square diagram corresponds to the graph
 
A square diagram corresponds to the graph
  
<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/d/d031/d031560/d03156033.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{lcl}
 +
i _ {1}  &\cdot \rightarrow ^ { {\phi _ 1} } \cdot  &i _ {2}  \\
 +
size - 3 {\phi _ {4} } \downarrow  &{}  &\downarrow size - 3 {\phi _ {2} }  \\
 +
i _ {4}  &\cdot  \mathop \rightarrow \limits _ { {\phi _ {3} }}  \cdot  &i _ {3}  \\
 +
\end{array}
 +
 
 +
$$
  
 
and is represented as follows:
 
and is represented 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/d/d031/d031560/d03156034.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{lcl}
 +
A _ {1}  &\rightarrow ^ { {f _ 1} }  &A _ {2}  \\
 +
size - 3 {f _ {4} } \downarrow  &{}  &\downarrow size - 3 {f _ {2} }  \\
 +
A _ {4}  & \mathop \rightarrow \limits _ { {f _ {3} }}  &A _ {3}  \\
 +
\end{array}
 +
 
 +
$$
  
Commutativity of this diagram means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156035.png" />.
+
Commutativity of this diagram means that $  f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4} $.
  
The class of diagrams with a given graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156036.png" /> forms a category. A morphism of a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156037.png" /> into a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156038.png" /> is taken to be a family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156040.png" /> runs through the set of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156041.png" />, so that for any edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156042.png" /> with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156043.png" /> and target <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156044.png" /> the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156045.png" /> is met. In particular, one may speak of isomorphic diagrams. The graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156046.png" /> is sometimes referred to as the scheme of a diagram in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031560/d03156047.png" />.
+
The class of diagrams with a given graph $  \Gamma $
 +
forms a category. A morphism of a diagram $  D $
 +
into a diagram $  D _ {1} $
 +
is taken to be a family of morphisms $  \nu _ {i} : D ( i) \rightarrow D _ {1} ( i) $,  
 +
where $  i $
 +
runs through the set of vertices of $  \Gamma $,  
 +
so that for any edge $  u $
 +
with source $  i $
 +
and target $  j $
 +
the condition $  D _ {1} ( u) \circ \nu _ {i} = \nu _ {j} \circ D ( u) $
 +
is met. In particular, one may speak of isomorphic diagrams. The graph $  \Gamma $
 +
is sometimes referred to as the scheme of a diagram in $  C $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR></table>

Revision as of 17:33, 5 June 2020


in a category $ C $

A mapping $ D $ of an oriented graph $ \Gamma $ with set of vertices $ I $ and set of edges $ U $ into the category $ C $ for which

$$ D ( I) \subset \mathop{\rm Ob} ( C) ,\ D ( U) \subset \mathop{\rm Mor} ( C) , $$

and $ D ( u) \in \mathop{\rm Hom} ( D ( i) , D ( j) ) $ if the edge $ u \in U $ has source (origin) $ i $ and target (end) $ j $. The concept of a diagram in $ C $ may also be defined as the image of the mapping $ D $, in order to obtain a better visualization of diagrams.

Let $ \phi = ( u _ {1} \dots u _ {n} ) $ be an oriented chain of the graph $ \Gamma $ with source $ i $ and target $ j $, i.e. a non-empty finite sequence of edges in which the source of each edge coincides with the target of the preceding one; also, let $ D ( \phi ) : D ( i) \rightarrow D ( j) $ denote the composition of morphisms

$$ D ( u _ {n} ) \circ \dots \circ D ( u _ {1} ) . $$

The diagram $ D $ is said to be commutative if $ D ( \phi ) = D ( \phi ^ \prime ) $ for any two oriented chains $ \phi $ and $ \phi ^ \prime $ with identical source and target.

The most frequently encountered forms of diagrams are sequences, triangular diagrams and square diagrams. To define a sequence, the defining graph is taken to have the form

$$ \cdot _ { i _ {1} } \rightarrow ^ { {u _ 1 } } \dots \rightarrow \cdot _ {i _ {n-} 1 } \rightarrow ^ { {u _ n-} 1 } \cdot _ {i _ {n} } . $$

The corresponding diagram is represented as follows:

$$ A _ {1} \rightarrow ^ { {f _ 1} } A _ {2} \rightarrow \dots \rightarrow A _ {n-} 1 \mathop \rightarrow \limits ^ { {f _ {n-} 1 }} A _ {n} , $$

where the $ A _ {k} = D ( i _ {k} ) $ are objects in the category $ C $, while the $ f _ {k} = D ( u _ {k} ) $ are morphisms in this category.

A triangular diagram in a category $ C $ corresponds to the graph

$$ \begin{array}{lcr} {} &i _ {1} & \mathop \rightarrow \limits _ { {u _ {3} }} \\ {} _ {u _ {1} } \nearrow &i _ {2} &\searrow _ {u _ {2} } \\ i _ {1} &{} \\ \end{array} $$

and is represented as follows:

$$ \begin{array}{lcr} A _ {1} &\rightarrow ^ { {f _ 1} } &A _ {2} \\ {} _ {f _ {3} } \searrow &{} &\swarrow _ {f _ {2} } \\ {} &A _ {3} &{} \\ \end{array} $$

Commutativity of this diagram means that $ f _ {3} = f _ {2} \circ f _ {1} $.

A square diagram corresponds to the graph

$$ \begin{array}{lcl} i _ {1} &\cdot \rightarrow ^ { {\phi _ 1} } \cdot &i _ {2} \\ size - 3 {\phi _ {4} } \downarrow &{} &\downarrow size - 3 {\phi _ {2} } \\ i _ {4} &\cdot \mathop \rightarrow \limits _ { {\phi _ {3} }} \cdot &i _ {3} \\ \end{array} $$

and is represented as follows:

$$ \begin{array}{lcl} A _ {1} &\rightarrow ^ { {f _ 1} } &A _ {2} \\ size - 3 {f _ {4} } \downarrow &{} &\downarrow size - 3 {f _ {2} } \\ A _ {4} & \mathop \rightarrow \limits _ { {f _ {3} }} &A _ {3} \\ \end{array} $$

Commutativity of this diagram means that $ f _ {2} \circ f _ {1} = f _ {3} \circ f _ {4} $.

The class of diagrams with a given graph $ \Gamma $ forms a category. A morphism of a diagram $ D $ into a diagram $ D _ {1} $ is taken to be a family of morphisms $ \nu _ {i} : D ( i) \rightarrow D _ {1} ( i) $, where $ i $ runs through the set of vertices of $ \Gamma $, so that for any edge $ u $ with source $ i $ and target $ j $ the condition $ D _ {1} ( u) \circ \nu _ {i} = \nu _ {j} \circ D ( u) $ is met. In particular, one may speak of isomorphic diagrams. The graph $ \Gamma $ is sometimes referred to as the scheme of a diagram in $ C $.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221

Comments

References

[a1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[a2] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
How to Cite This Entry:
Diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram&oldid=46644
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article