Quotient of an abelian category information

In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category A {\displaystyle {\mathcal {A}}} by a Serre subcategory...

original category. The Serre quotient of an abelian category by a Serre subcategory is a new abelian category which is similar to a quotient category but also...

prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are...

mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every...

such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f : X → Y is the quotient of Y by the image of f. In topological...

Serre subcategory B, one can define the quotient category A/B, which is an abelian category equipped with an exact functor from A to A/B that is essentially...

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...

One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically...

gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly...

also refer to a category with a monoidal-category action. The categories of left and right modules are abelian categories. These categories have enough projectives...

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is...

covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup...

terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory...

Ab, is a full subcategory of Grp. Ab is an abelian category, but Grp is not. Indeed, Grp isn't even an additive category, because there is no natural...

In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It...

subalgebra is an ideal in g {\displaystyle {\mathfrak {g}}} , in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous...

specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups...

(the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra...

Category of preordered sets Category of groups Category of abelian groups Category of rings Category of magmas Initial object Terminal object Zero object Subobject...

the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulated category, and...

particular, any non-trivial category with a zero object, such as an abelian category, is not Cartesian closed. So the category of modules over a ring is not...

many categories based on structured sets. See free object. Homomorphism groups To every pair A, B of abelian groups one can assign the abelian group...

from the category of abelian groups to category of groups. It has a left adjoint called abelianization which assigns to every group G the quotient group...

is that an abelian group G {\displaystyle G} is divisible if and only if G {\displaystyle G} is an injective object in the category of abelian groups;...