PlanetMath: groupoid homomorphism

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groupoid homomorphism

(Definition)

Definition 0.1 Let ${\mathsf{\mathcal G}}_1$ and ${\mathsf{\mathcal G}}_2$ be two groupoids considered as two distinct categories with all invertible morphisms between their objects (or `elements'), respectively, $x \in Ob({\mathsf{\mathcal G}}_1) = {{{\mathsf{\mathcal G}}_0}}^1$ and $y \in Ob({\mathsf{\mathcal G}}_2) = {{{\mathsf{\mathcal G}}_0}}^2$ . A groupoid homomorphism is then defined as a functor $h: {\mathsf{\mathcal G}}_1 \longrightarrow {\mathsf{\mathcal G}}_2$ .

A composition of groupoid homomorphisms is naturally a homomorphism, and natural transformations of groupoid homomorphisms (as defined above by groupoid functors) preserve groupoid structure(s), i.e., both the algebraic and the topological structure of groupoids. Thus, in the case of topological groupoids, $\mathsf{G}$ , one also has the associated topological space homeomorphisms that naturally preserve topological structure.

Remark: Note that the morphisms in the category of groupoids, , are, of course, groupoid homomorphisms, and that groupoid homomorphisms also form (groupoid) functor categories defined in the standard manner for categories.

"groupoid homomorphism" is owned by bci1.

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Other names:

groupoid morphisms

Also defines:

morphism of groupoids

Keywords:

homomorphism of groupoids, groupoid category, morphisms of groupoids

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Cross-references: functor categories, category of groupoids, homeomorphisms, topological space, algebraic, structure, preserve, natural transformations, homomorphism, composition, functor, objects, morphisms, invertible, categories, groupoids
There are 9 references to this entry.

This is version 29 of groupoid homomorphism, born on 2008-07-28, modified 2008-09-04.
Object id is 10888, canonical name is GroupoidHomomorphisms.
Accessed 645 times total.

Classification:

AMS MSC:	18-00 (Category theory; homological algebra :: General reference works )
	18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories)
	55U35 (Algebraic topology :: Applied homological algebra and category theory :: Abstract and axiomatic homotopy theory)
	55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
	18D35 (Category theory; homological algebra :: Categories with structure :: Structured objects in a category )

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