Abelian category

Definition

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

Related concepts

0 (number)2-functor 2-group 2-ring 3-category AB5 category Abelian group Academic Press Additive category Additive functor Adjoint functors Alexander Grothendieck Algebraic category Algebraic geometry Algebraic topology American Mathematical Society Axiom Bicategory Bilinear operator Biproduct Bounded lattice Cartesian closed category Categories for the Working Mathematician Categorification Category (mathematics)Category of abelian groups Category of groups Category of magmas Category of modules Category of relations Category of rings Category of sets Category theory Chain complex Class (set theory)Cocomplete Coequalizer Coherent sheaf Cohomological dimension Cohomology Coimage Cokernel Colimit Comma category Commutative algebra Commutative diagram Commutative ring Comodule Complete category Concrete category Cone (category theory)Conservative functor Coproduct David Buchsbaum David Eisenbud Derived functor Diagonal functor Diagram (category theory)Direct limit Direct product Direct sum Direct sum of modules En-ring End (category theory)Enriched category Enriched limit Epimorphism Equaliser (mathematics)Equivalence of categories Essentially surjective functor Exact category Exact functor Exact sequence Exponential object Field (mathematics)Filtered colimit Filtered limit Finitary Finite set Finitely generated abelian group Finitely generated module Five lemma Flat module Flat morphism Forgetful functor Free category Full and faithful functors Full subcategory Functor Functor category G-module Glossary of category theory Graduate Texts in Mathematics Grothendieck topology Group (mathematics)Hamel dimension Higher-dimensional algebra Higher category theory Hom-set Homological algebra Homotopy hypothesis Image (category theory)Initial and terminal objects Inverse limit Isomorphism Isomorphism of categories Kan extension Kernel (category theory)Kleisli category Lie group Limit (category theory)Local algebra Localization of a category Localizing subcategory Maschke's theorem Mathematics Mitchell's embedding theorem Model category Module (mathematics)Monoidal category Monoidal functor Monomorphism Morphism N-group (category theory)N-monoid Natural transformation Nicolae Popescu Nine lemma Noetherian ring Noetherian scheme Normal morphism Object (category theory)Opposite category Outline of category theory Overcategory Peter Freyd Poset Pre-abelian category Preabelian category Preadditive category Product (category theory)Product category Pseudo-functor Pullback (category theory)Pushout (category theory)Quasi-category Quotient category Quotient object Regular category Representable functor Representation theory Ring (mathematics)Samuel Eilenberg Saunders Mac Lane Scheme (mathematics)Semi-simplicity Semisimple ring Sheaf (mathematics)Short exact sequence Short five lemma Simplex category Small category Smooth functor Snake lemma Strict 2-category Strict n-category String diagram Subcategory Subobject Subquotient Symmetric monoidal category Tensor product Tetracategory Tohoku Mathematical Journal Topological space Topos Traced monoidal category Transactions of the American Mathematical Society Triangulated category Tricategory Tōhoku article Unipotent group Universal construction Universal property Vector bundles Vector space Weak n-category Well-behaved Yoneda lemma Zariski tangent space Zero morphism Zero object ∞-groupoid ∞-topos

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