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- 14:55, 29 December 2009 Abelian group (hist) [197 bytes] Vipul (Talk | contribs) (Created page with '==Definition== An '''abelian group''' is a group in which any two elements commute. The collection of abelian groups is often studied as a category, called the [[category oâ€¦')
- 14:50, 29 December 2009 2-category (hist) [4,437 bytes] Vipul (Talk | contribs) (Created page with '==Definition== A '''2-category''' <math>\mathcal{C}</math> is the following data: * '''Objects''': A defining ingredient::collection <math>\operatorname{Ob}\mathcal{C}</matâ€¦')
- 12:51, 29 December 2009 Group (hist) [2,296 bytes] Vipul (Talk | contribs) (Created page with '==Definition== ===Category-theoretic definition=== A '''group''' is a small category with one object where all morphisms are isomorphisms. (note that the ''small'' assumptiâ€¦')
- 07:15, 26 December 2008 Category of algebras in a variety (hist) [1,745 bytes] Vipul (Talk | contribs) (New page: ==Definition== Let <math>\mathcal{V}</math> be a variety of algebras. In other words, <math>\mathcal{V}</math> is the collection of all algebras with a particular operator domain (eac...)
- 06:52, 26 December 2008 Abelian category (hist) [1,076 bytes] Vipul (Talk | contribs) (New page: {{preadditive category property}} ==Definition== An Abelian category is a category enriched over the monoidal category of Abelian groups satisfying the ...)
- 06:43, 26 December 2008 Preabelian category (hist) [1,124 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===In terms of additive category=== A '''preabelian category''' (sometimes written '''preAbelian category''' or '''pre-Abelian category''') is an [[defining ingredient::ad...)
- 06:35, 26 December 2008 Conormal category (hist) [202 bytes] Vipul (Talk | contribs) (New page: ==Definition== A '''conormal category''' is a category enriched over the monoidal category of pointed sets in which every epimorphism is [[normal epimorp...)
- 06:34, 26 December 2008 Normal category (hist) [513 bytes] Vipul (Talk | contribs) (New page: ==Definition== A '''normal category''' is a category enriched over the monoidal category of pointed sets satisfying the condition that every monomorphism...)
- 05:04, 26 December 2008 Normal epimorphism (hist) [1,882 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the monoidal category of Abelian groups), a '''norm...)
- 12:16, 25 December 2008 Normal monomorphism (hist) [2,173 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===In a preadditive category=== In a preadditive category (i.e., a category enriched over the category of Abelian groups), a '''normal monomo...)
- 12:10, 25 December 2008 Biproduct (hist) [1,433 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===For two objects=== Suppose <math>\mathcal{C}</math> is a category and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''biproduct''' of <math>A_1</math> a...)
- 11:41, 25 December 2008 Category of small categories (hist) [643 bytes] Vipul (Talk | contribs) (New page: {{particular category}} ==Definition== The '''category of small categories''', sometimes denoted <math>\operatorname{Cat}</math>, is defined as follows: * The objects of the category ar...)
- 01:44, 10 December 2008 Additive category (hist) [390 bytes] Vipul (Talk | contribs) (New page: {{category plus additional structure}} ==Definition== An '''additive category''' is defined as a preadditive category that also admits finite products and coproducts. ==Rela...)
- 01:43, 10 December 2008 Preadditive category (hist) [488 bytes] Vipul (Talk | contribs) (New page: {{category plus additional structure}} ==Definition== A '''preadditive category''' is a category enriched over the monoidal category of Abelian groups. Note tha...)
- 01:40, 10 December 2008 Monoidal category of abelian groups (hist) [325 bytes] Vipul (Talk | contribs) (New page: {{particular monoidal category}} ==Definition== The '''monoidal category of Abelian groups''' is defined as follows: * The underlying category is the category of Abelian groups. * T...) originally created as "Monoidal category of Abelian groups"
- 01:39, 10 December 2008 Category of abelian groups (hist) [963 bytes] Vipul (Talk | contribs) (New page: {{particular category}} ==Definition== The '''category of Abelian groups''', sometimes denoted <math>\operatorname{Ab}</math>, is defined as follows: * Its objects are Abelian group...) originally created as "Category of Abelian groups"
- 01:22, 10 December 2008 Locally finitely presentable category (hist) [666 bytes] Vipul (Talk | contribs) (New page: ==Definition== A '''locally finitely presentable category''' is a locally small category that satisfies the following conditions: * It is [[defining ingredient::cocomplete category|c...)
- 01:09, 10 December 2008 Pullback-preserving functor (hist) [257 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor between two categories is termed '''pullback-preserving''' if it maps any [[pullback diagram]...)
- 01:07, 10 December 2008 Product-preserving functor (hist) [961 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor between two categories is termed '''product-preserving''' if it preserves [[categorical produc...)
- 00:58, 10 December 2008 Enriched category (hist) [1,496 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{M}</math> is a defining ingredient::monoidal category. A '''category enriched in''' <math>\mathcal{M}</math> is defined as a <math>\mathcal{C}</m...)
- 00:05, 10 December 2008 Category of pointed sets (hist) [385 bytes] Vipul (Talk | contribs) (New page: {{particular category}} ==Definition== The '''category of pointed sets''', denoted <math>\operatorname{Set}_*</math>, is defined as followed: * Its objects are pointed sets. A pointed s...)
- 23:56, 9 December 2008 Category of topological spaces (hist) [511 bytes] Vipul (Talk | contribs) (New page: {{particular category}} ==Definition== The '''category of topological spaces''', denoted <math>\operatorname{Top}</math>, is defined as follows: * Its objects are topological spaces...)
- 23:53, 9 December 2008 Category of groups (hist) [543 bytes] Vipul (Talk | contribs) (New page: {{particular category}} ==Definition== The '''category of groups''', denoted <math>\operatorname{Group}</math> or <math>\operatorname{Grp}</math>, is defined as follows: * Its objects a...)
- 23:44, 9 December 2008 Category of sets (hist) [1,537 bytes] Vipul (Talk | contribs) (New page: ==Definition== The '''category of sets''', denoted <math>\operatorname{Set}</math>, is defined as follows: * The objects of this category are sets. * For any two sets <math>A,B</math...)
- 23:42, 9 December 2008 Natural isomorphism (hist) [1,001 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} ==Definition== ===Symbol-free definition=== A '''natural isomorphism''' between functors is a natural transformation between the functors that has a two-sided inverse w...)
- 23:36, 9 December 2008 Representable contravariant functor (hist) [529 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C}</math> is a category and <math>\mathcal{F}:\mathcal{C} \to \operatorname{Set}</math> is a contravariant functor to the [[category of sets]...)
- 23:33, 9 December 2008 Covariant Hom functor (hist) [1,208 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C}</math> is a category and <math>A \in \operatorname{Ob}\mathcal{C}</math>. The '''covariant Hom functor''' corresponding to <math>A</math> is d...)
- 23:28, 9 December 2008 Representable functor (hist) [627 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C}</math> is a category, and <math>\mathcal{F}: \mathcal{C} \to \operatorname{Set}</math> is a functor. In other words, <math>\mathcal{F}</ma...)
- 23:17, 9 December 2008 Contravariant functor (hist) [1,671 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===Definition in basic terms=== Suppose <math>\mathcal{C}, \mathcal{D}</math> are categories. A '''contravariant functor''' <math>\mathcal{F}:\mathcal{C} \to ...)
- 02:04, 9 December 2008 Cocomplete category (hist) [268 bytes] Vipul (Talk | contribs) (New page: {{category property}} ==Definition== A '''cocomplete category''' is a category in which all small colimits exist. In particular, it admits coproducts and coequalizers. ==Rel...)
- 02:03, 9 December 2008 Complete category (hist) [299 bytes] Vipul (Talk | contribs) (New page: {{category property}} ==Definition== A '''complete category''' is a category that admits all small limits. In particular, it admits all categorical products, direct limit...)
- 01:56, 9 December 2008 Symmetric monoidal category (hist) [514 bytes] Vipul (Talk | contribs) (New page: {{category plus additional structure}} ==Definition== A '''symmetric monoidal category''' is a monoidal category <math>(\mathcal{C}, \otimes)</math> equipped with a commutativity iso...)
- 01:52, 9 December 2008 Monoidal category (hist) [677 bytes] Vipul (Talk | contribs) (New page: {{category plus additional structure}} ==Definition== A '''monoidal category''' is a category <math>\mathcal{C}</math> equipped with an additional operation, called a monoidal operat...)
- 01:32, 9 December 2008 Opposite category (hist) [781 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C}</math> is a category. The '''opposite category''' to <math>\mathcal{C}</math>, denoted <math>\mathcal{C}^{op}</math>, is defined as ...)
- 01:29, 9 December 2008 Adjoint functors (hist) [782 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C}</math> and <math>\mathcal{D}</math> are categories. Suppose <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> and <math>\mathcal{G...)
- 01:24, 9 December 2008 Zero object (hist) [1,041 bytes] Vipul (Talk | contribs) (New page: ==Definition== An object in a category is termed a '''zero object''' if it is both an defining ingredient::initial object and a defining ingredient::final object.)
- 01:21, 9 December 2008 Categorical product (hist) [864 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C}</math> is a category and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''categorical product''', or simply '''product''', of <math>A_1...)
- 01:17, 9 December 2008 Final object (hist) [545 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} ==Definition== ===Definition with symbols=== An object <math>A</math> in a category <math>\mathcal{C}</math> is termed a '''final object''' if for every object <math>B ...)
- 01:14, 9 December 2008 Equivalence of categories (hist) [605 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== An '''equivalence of categories]] is a functor between two categories that satisfies the following thre...)
- 01:12, 9 December 2008 Essentially surjective functor (hist) [555 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} {{functor property}} ==Definition== ===Symbol-free definition=== A functor is termed '''essentially surjective''' if every object in the target category is isomorph...)
- 01:08, 9 December 2008 Initial object (hist) [1,337 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C}</math> is a category. An '''initial object''' in <math>\mathcal{C}</math> is an object <math>A \in \operatorname{Ob}\mathcal{C}</mat...)
- 00:58, 9 December 2008 Natural transformation (hist) [730 bytes] Vipul (Talk | contribs) (New page: {{basicdef}} ==Definition== Suppose <math>\mathcal{C},\mathcal{D}</math> are categories, and <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> and <math>\mathcal{G}:\math...)
- 00:53, 9 December 2008 Epimorphism (hist) [551 bytes] Vipul (Talk | contribs) (New page: {{morphism property}} ==Definition== Suppose <math>\mathcal{C}</math> is a category, <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B)</math>. <math>f</...)
- 00:50, 9 December 2008 Monomorphism (hist) [621 bytes] Vipul (Talk | contribs) (New page: ==Definition== ===Definition with symbols=== Suppose <math>\mathcal{C}</math> is a category, <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B)</math...)
- 00:43, 9 December 2008 Concrete category (hist) [324 bytes] Vipul (Talk | contribs) (New page: {{category with additional structure}} ==Definition== A '''concrete category''' is the data of a defining ingredient::category along with a defining ingredient::faithful functor ...)
- 00:41, 9 December 2008 Full functor (hist) [680 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== A '''full functor''' is a functor such that the induced mapping on the collection of morphisms between any two objec...)
- 00:37, 9 December 2008 Conservative functor (hist) [783 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor is termed '''conservative''' if it reflects isomorphisms: in other words, any morphism that gets mapped to...)
- 00:27, 9 December 2008 Small category (hist) [346 bytes] Vipul (Talk | contribs) (New page: {{category property}} ==Definition== A '''small category''' is a category satisfying ''both'' the following: * The collection of objects for the category is a set. * For any two objects...)
- 00:14, 9 December 2008 Faithful functor (hist) [550 bytes] Vipul (Talk | contribs) (New page: {{functor property}} ==Definition== ===Symbol-free definition=== A functor is faithful if the induced map on the collection of morphisms between any pair of objects is injective. ===Def...)
- 00:12, 9 December 2008 Functor (hist) [1,210 bytes] Vipul (Talk | contribs) (New page: ==Definition== Suppose <math>\mathcal{C},\mathcal{D}</math> are categories. A '''functor''' <math>\mathcal{F}</math> from <math>\mathcal{C}</math> to <math>\mathcal{D}</math>...)