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Longer titles found: Pre-abelian category (view), Pseudo-abelian category (view), Quotient of an abelian category (view), Semi-abelian category (view), Quasi-abelian category (view)

searching for Abelian category 9 found (155 total)

alternate case: abelian category

Kernel (category theory) (950 words) [view diff] exact match in snippet view article

always exists in an abelian category) turns out to be the image of that morphism; in symbols: im f = ker coker f (in an abelian category) When m is a monomorphism

Exact functor (2,404 words) [view diff] exact match in snippet view article find links to article

basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant

Representation theorem (608 words) [view diff] exact match in snippet view article find links to article

Mitchell's embedding theorem for abelian categories realises every small abelian category as a full (and exactly embedded) subcategory of a category of modules

Weak equivalence (homotopy theory) (868 words) [view diff] exact match in snippet view article

model categories involve chain complexes. Let A be a Grothendieck abelian category, for example the category of modules over a ring or the category of

Brown's representability theorem (845 words) [view diff] exact match in snippet view article find links to article

complexes, as well as to the unbounded derived category of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category)

Homotopy category (1,723 words) [view diff] exact match in snippet view article find links to article

more algebraic example of a model category, let A be a Grothendieck abelian category, for example the category of modules over a ring or the category of

Differential graded category (828 words) [view diff] exact match in snippet view article find links to article

do exist, for example the derived category D(A) of a Grothendieck abelian category A admits a unique dg enhancement. Differential algebra Graded (mathematics)

Tilting theory (1,690 words) [view diff] exact match in snippet view article find links to article

construction. Colpi & Fuller (2007) defined tilting objects T in an arbitrary abelian category C; their definition requires that C contain the direct sums of arbitrary

Grothendieck spectral sequence (1,590 words) [view diff] exact match in snippet view article find links to article

use the following lemma: Lemma — If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects