Pre-abelian category (original) (raw)
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
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| dbo:abstract | En mathématiques, plus précisément en théorie des catégories, une catégorie préabélienne est une catégorie additive qui contient tous les noyaux et conoyaux. De manière plus détaillée, cela signifie qu'une catégorie C est pré-abélienne si: 1. * C est , c'est-à-dire enrichie sur une catégorie monoïdale de groupes abéliens (de manière équivalente, toutes les collections de morphismes d'un objet de C vers un objet de C sont des groupes abéliens et une composition de morphismes est bilinéaire) 2. * C contient tous les produits finis (de manière équivalente, tous les coproduits finis). On notera que, comme C est aussi préadditive, les produits finis sont les mêmes que les coproduits finis 3. * étant donné que tout morphisme f: A → B dans C, l'égaliseur de f, et que le morphisme zéro de A à B existe (par définition, c'est le noyau de f), ainsi que le coégaliseur (qui est par définition le conoyau de f). On notera que le morphisme zéro au point 3 peut être identifié à l'élément neutre de l'ensemble des morphismes de A vers B Hom(A,B), qui est, d'après le point 1, un groupe abélien ou au morphisme unique A → O → B, où O est un objet zéro, dont l'existence est garantie par le point 2. (fr) In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: 1. * C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); 2. * C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; 3. * given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2. (en) Em matemática, especificamente em teoria das categorias, uma categoria pré-abeliana é uma que tem todos os e . Tratando em mais detalhes, isto significa que uma categoria C é pré-Abeliana se: 1. * C é , que é sobre a de grupos abelianos; 2. * C tem todos os , os quais são todos finitos e coprodutos finitos; 3. * dado qualquer morfismo f: A → B em C, o equalizador de f e o de A a B existe (este é o núcleo), como o coequalizador (este é o conúcleo). Note-se que o morfismo zero no item 3 pode ser identificado como o elemento identidade do conjunto hom Hom(A,B), o qual é um grupo Abeliano pelo item 1; ou como o único morfismo A → O → B, onde O é um , guarantido como existente pelo item 2. (pt) |
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| rdfs:comment | In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2. (en) En mathématiques, plus précisément en théorie des catégories, une catégorie préabélienne est une catégorie additive qui contient tous les noyaux et conoyaux. De manière plus détaillée, cela signifie qu'une catégorie C est pré-abélienne si: On notera que le morphisme zéro au point 3 peut être identifié à l'élément neutre de l'ensemble des morphismes de A vers B Hom(A,B), qui est, d'après le point 1, un groupe abélien ou au morphisme unique A → O → B, où O est un objet zéro, dont l'existence est garantie par le point 2. (fr) Em matemática, especificamente em teoria das categorias, uma categoria pré-abeliana é uma que tem todos os e . Tratando em mais detalhes, isto significa que uma categoria C é pré-Abeliana se: 1. * C é , que é sobre a de grupos abelianos; 2. * C tem todos os , os quais são todos finitos e coprodutos finitos; 3. * dado qualquer morfismo f: A → B em C, o equalizador de f e o de A a B existe (este é o núcleo), como o coequalizador (este é o conúcleo). (pt) |
| rdfs:label | Catégorie préabélienne (fr) Pre-abelian category (en) Categoria pré-abeliana (pt) |
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