...t a formal axiomatic theory, obtained within a definite [[Meta-theory|meta-theory]].
...imbedded (in a structure-preserving way) into (a power of) the particular category under consideration.
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...n every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual
[[Category:Category theory; homological algebra]]
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...ace $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.)
[[Category|category]] of open sets of $X$ and their natural inclusion mappings into $\cK$. Depe
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...nother. Two categories are equivalent if and only if their [[Skeleton of a category|skeletons]] are isomorphic.
...ts (cf. the editorial comments to [[Category]] for the notion of a Kleisli category of a triple).
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A [[subcategory]] $\mathfrak C$ of a [[category]] $\mathfrak K$ such that for any objects $A$ and $B$ from $\mathfrak C$ on
...lass of its objects. Conversely, any subclass of the class of objects of a category $\mathfrak K$ uniquely defines a full subcategory, for which it serves as t
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A category $\mathfrak K$ in which subcategories of epimorphisms $\mathfrak E$ and of m
...mathfrak E\cap\mathfrak M$ coincides with the class of isomorphisms in the category $\mathfrak R$.
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''of a family of objects in a category''
...t of a family of objects in a category|product of a family of objects in a category]].
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''in a category''
...t). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$,
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''of a category''
...ects and the class of morphisms, respectively. The class of morphisms of a category $\mathfrak{K}$ is usually denoted by $\operatorname{Mor} \mathfrak{K}$.
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...rns out to be the kernel of its cokernel. In an [[Abelian category|Abelian category]] every monomorphism is normal. The concept of a normal monomorphism is dua
...isomorphism of $G$ onto a normal subgroup of $H$. However, in an additive category the concepts of normal monomorphism and regular monomorphism coincide.
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''category of sequences''
...relation. Then $\mathbb{Z}$ can be considered as a [[Small category|small category]] with integers as objects and all possible pairs $(i,j)$, where $i,j \in \
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...replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...[[Closed monoidal category|closed monoidal category]] (cf. also [[Category|Category]]). A [[Functor|functor]] $( - ) ^ { * } : \cal C ^ { \operatorname{op} } \
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...f a category|Null object of a category]]). An axiomatic description of the category of groups was given by P. Leroux [[#References|[3]]].
...oup object|Group object]]) in $K$ and the homomorphisms between them; this category has some of the properties of $K$; in particular, it is complete if $K$ is
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[[Category:Topology]]
...set|nowhere dense sets]] in $X$, otherwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).
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$#C+1 = 28 : ~/encyclopedia/old_files/data/Q076/Q.0706870 Quotient category
be an arbitrary [[Category|category]], and suppose that an equivalence relation $ \sim $
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''terminal object, of a category''
...ight null object of $\mathfrak{K}$. A left null or ''initial object'' of a category is defined in the dual way.
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An [[Abelian category]] with a set of generators (cf. [[Generator of a category]]) and satisfying the following axiom: There exist [[coproduct]]s (sums) of
...y]]) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a ''localizing subcateg
2 KB (366 words) - 19:42, 30 October 2016
A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group str
...null object (zero object, cf. [[Null object of a category|Null object of a category]]) as well as the product $X\times Y$ of any two objects $X$ and $Y$.
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A [[category]] $\mathfrak{C}$ such that the following axioms are satisfied:
These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors
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...e subsets often do not); see [[#References|[a1]]], for example. In lattice theory, least upper bounds of directed subsets again play a distinctive part; see
[[Category:Order, lattices, ordered algebraic structures]]
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...ral transformations, the category of modules over $\Gamma$ is an [[Abelian category]], so one can do [[homological algebra]] with these objects.
...ilarly, equivariant local cohomology can be described using modules over a category depending on the space in question.
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[[Category:Linear and multilinear algebra; matrix theory]]
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...re all sets belonging to $U$, with morphisms and composition as above. The category of sets may be denoted by $\mathfrak S$, ENS, $\mathsf{Set}$ or Me.
...t every epimorphism is split is equivalent to the [[axiom of choice]]. The category of sets has a unique [[Bicategory(2)|bicategory]] (factorization) structure
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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A [[category]] with an additional structure, thanks to which the internal Hom-functor ca
A category $\mathfrak{M}$ is said to be closed if a [[bifunctor]] $\otimes: \mathfrak{
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...}}(Y,X)$ defines a contravariant functor $h_X$ from $\mathcal{C}$ into the category of sets. For any object $F$ of $\hat{\mathcal{C}}$ there exists a natural b
...ieck functor it is possible to define algebraic structures on objects of a category (cf. [[Group object]]; [[Group scheme]]).
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[[Category:Number theory]]
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$#C+1 = 35 : ~/encyclopedia/old_files/data/M064/M.0604480 Modules, category of
The [[Category|category]] mod- $ R $
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[[Category:Number theory]]
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[[Category:Classical measure theory]]
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[[Category:Classical measure theory]]
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...pological vector space which is not a set of the [[Category of a set|first category]] is ultra-barrelled. If a [[locally convex space]] is ultra-barrelled, it
...gn="top">[1]</TD> <TD valign="top"> R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)</TD></TR>
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...system with multiple inputs and multiple outputs; see [[Automatic control theory]].
[[Category:Control theory and optimization]]
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...an 4 cannot, in general, be solved by radicals (see [[Galois theory|Galois theory]]).
...Many questions of the theory of radicals have been studied within category theory. See also [[Radical of a group|Radical of a group]]; [[Radical in a class o
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...lexes or simplicial decompositions. Simplicial spaces are the objects of a category whose morphisms $X\to Y$ are mappings such that every simplex of the triang
...gical spaces (cf. [[Simplicial object in a category|Simplicial object in a category]]).
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[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
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...s, the exponential law makes the [[category of sets]] a [[Cartesian-closed category]].
* Benjamin C. Pierce, ''Basic Category Theory for Computer Scientists'', MIT Press (1991) {{ISBN|0262660717}}
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[[Category:Group theory and generalizations]]
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...kernel of a homomorphism of groups, rings, etc. Let $\mathfrak{K}$ be a [[category]] with zero or [[null morphism]]s. A morphism $\mu : K \to A$ is called a k
...$ contains a null object (cf. [[Null object of a category|Null object of a category]]).
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[[Category:Classical measure theory]]
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...reflective if it contains a reflection (cf. [[Reflection of an object of a category]]) for every object of $\mathfrak{K}$. Equivalently, $\mathfrak{C}$ is refl
...$\mathfrak{C}$. Thus, a reflective subcategory of a complete (cocomplete) category is complete (cocomplete).
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The ''fibre product of objects in a category'' is
...t|(inverse or projective) limit]]. Let $\def\fK{ {\mathfrak K}}\fK$ be a [[category]] and let $\def\a{\alpha}\a : A\to C$ and $\def\b{\beta}\b : B\to C$ be giv
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...mappings of sets. A [[Morphism|morphism]] $\pi : A \to B$ in a [[Category|category]] $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$
...ct of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorn
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See also [[Duality|Duality]] in the theory of [[topological vector space]]s.
[[Category:Linear and multilinear algebra; matrix theory]]
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A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data:
1) a category $\mathcal{C}$;
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...oherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.
gives rise to an equivalence of the category of quasi-coherent sheaves of $ {\mathcal A} $-
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...of several arguments, defined on categories, taking values in a [[Category|category]] and giving a one-place [[Functor|functor]] in each argument. More precise
be given. Construct the Cartesian product category $ \mathfrak K = \overline{\mathfrak K}\; _ {1} \times \dots \times \overl
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...between ($\mathcal{U}$-) categories, and in order to admit other "large" category-theoretic constructions.
...olland (1977) ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR>
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[[Category:Linear and multilinear algebra; matrix theory]]
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