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An additive category with kernels and cokernels in which monomorphisms and epimorphisms are normal.
wp-abelian-categorynlab-abelian-categorystacks-abelian-categoryriehl-category-contextEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require kernels and cokernels and normality of monomorphisms and epimorphisms.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
nlab-abelian-categorystacks-abelian-categoryriehl-category-contextEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Form complexes in , pass to the homotopy category , and localize at quasi-isomorphisms. The result is a new category, not an equivalent presentation of .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
stacks-derived-categorystacks-derived-moduleswp-derived-categorynlab-derived-categorystacks-derived-textThis is an authored directed relation from the source endpoint to the target endpoint.
Every abelian category has a canonical Quillen exact structure consisting of all short exact sequences.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
nlab-abelian-categorystacks-abelian-categorywp-exact-categorynlab-exact-categorystacks-derived-textThis is an authored directed relation from the source endpoint to the target endpoint.
Require exact filtered colimits, small coproducts, and a generator.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-grothendieck-categorynlab-grothendieck-categorystacks-derived-text