...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.
1 KB (173 words) - 17:22, 7 February 2011
[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]
264 bytes (38 words) - 07:19, 19 September 2012
...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]]
941 bytes (147 words) - 21:29, 1 November 2014
...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
850 bytes (133 words) - 16:46, 24 November 2013
...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).
1 KB (231 words) - 07:37, 28 November 2017
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
1 KB (160 words) - 17:51, 15 November 2014
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 $\mathfrak R$.
2 KB (267 words) - 10:09, 23 August 2014
''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]].
6 KB (867 words) - 13:57, 26 December 2017
''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 $\mu$,
2 KB (279 words) - 05:35, 12 January 2017
''of a category''
...ects and the class of morphisms, respectively. The class of morphisms of a category $\mathfrak{K}$ is usually denoted by $\operatorname{Mor} \mathfrak{K}$.
2 KB (284 words) - 13:56, 26 December 2017
...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.
2 KB (314 words) - 02:26, 14 January 2017
''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 \
2 KB (380 words) - 11:48, 26 October 2014
...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} } \
3 KB (375 words) - 17:46, 1 July 2020
...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
3 KB (379 words) - 05:17, 12 January 2017
[[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}}).
2 KB (291 words) - 19:06, 7 December 2023
$#C+1 = 28 : ~/encyclopedia/old_files/data/Q076/Q.0706870 Quotient category
be an arbitrary [[Category|category]], and suppose that an equivalence relation $ \sim $
2 KB (279 words) - 08:09, 6 June 2020
''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.
2 KB (322 words) - 21:19, 21 December 2017
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 $Y$.
3 KB (490 words) - 23:53, 10 December 2018
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
2 KB (374 words) - 20:31, 27 December 2017