PlanetPhysics/Categories of Groupoids

Groupoid categories , or categories of groupoids , can be defined simply by considering a groupoid as a category {Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G}_1} } with all invertible morphisms, and objects defined by the groupoid class or set of groupoid elements; then, the groupoid category, Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _2} }, is defined as the -category whose objects are Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _1} } categories (groupoids), and whose morphisms are functors of Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _1} } categories consistent with the definition of groupoid homomorphisms, or in the case of topological groupoids, consistent as well with topological groupoid homeomorphisms. The 2-category of groupoids Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _2} }, plays a central role in the generalised, categorical Galois theory involving fundamental groupoid functors.

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