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examples of abelian categories
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(Example)
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The axiomatization of abelian categories was intended to capture some of the useful properties of categories in homological algebra. This entry gives some examples of abelian categories.
Since abelian groups are a special case of $R$ -modules, one might ask whether categories of $R$ -modules are also abelian categories. This is the case.
Example 2 For any ring $R$ , the category $\Mod{R}$ of left $R$ -modules is an abelian category.
Example 3 For any ring $R$ , the category $\mathcal{C}(R)$ of complexes of left $R$ -modules is an abelian category.
Example 7 The 2-category of commutative groupoids $\mathcal{G_C}$ is an Abelian 2-category.
Example 8 For any reversible sequential machine, or automaton, $R_S$ , (with all state transitions reversible), the category $\mathcal{A}(R_S)$ of such reversible automata is an Abelian category.
Counter-example For general quantum automata $Q_A$ s, the category $\mathcal{Q_A}$ of such quantum automata is in general, non-Abelian (or nonabelian).
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"examples of abelian categories" is owned by mps. [ full author list (2) ]
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See Also: Grothendieck category, non-Abelian structures, quantum automata and computation, abelian category, supplemental axioms for an Abelian category, higher dimensional generalized Van Kampen theorems (HD-VKT), axiomatic theory of supercategories and metacategories, algebraic category of LMn logic algebras, categorical quantum logics as quantum LM-algebraic logic, non-commuting graph, non-Abelian structures, proof that abelian groups form an abelian category, index of categories
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Abelian groups, Grothendieck categories, commutative groupoids, reversible automata, quantum automata, rings, modules |
This object's parent.
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Cross-references: non-Abelian, automata, automaton, sequential machine, 2-category, groupoid, groupoids, commutative, topological groupoid, abelian, axiom, Grothendieck category, sheaves, topological space, ring, abelian groups, algebra, categories, useful, abelian categories
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This is version 25 of examples of abelian categories, born on 2007-02-20, modified 2008-10-28.
Object id is 8931, canonical name is ExamplesOfAbelianCategory.
Accessed 1236 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) |
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Pending Errata and Addenda
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