PlanetPhysics/Categories of Groupoids

Groupoid categories , or categories of groupoids , can be defined simply by considering a [[../GroupoidHomomorphism2/|groupoid]] as a [[../Cod/|category]] {Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G}_1} } with all invertible [[../TrivialGroupoid/|morphisms]], and [[../TrivialGroupoid/|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 $2$-category whose objects are Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _1} } categories (groupoids), and whose morphisms are [[../TrivialGroupoid/|functors]] of Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G _1} } categories consistent with the definition of [[../GroupoidHomomorphism2/|groupoid homomorphisms]], or in the case of [[../GroupoidHomomorphism2/|topological groupoids]], consistent as well with topological groupoid homeomorphisms. The [[../2Category/|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 [[../QuantumFundamentalGroupoid3/|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 [[../Cod/|composition]] of groupoid homomorphisms is naturally a [[../TrivialGroupoid/|homomorphism]], and [[../VariableCategory2/|natural transformations]] of groupoid homomorphisms (as defined above by [[../GroupoidHomomorphism/|groupoid functors]]) preserve groupoid structure(s), i.e., both the [[../CoIntersections/|algebraic]] and the [[../TrivialGroupoid/|topological structure]] of groupoids. Thus, in the case of topological groupoids, $𝖦$, one also has the associated [[../CoIntersections/|topological]] space [[../TrivialGroupoid/|homeomorphisms]] that naturally preserve topological structure.

Remark: Note that the morphisms in the [[../GroupoidCategory/|category of groupoids]], $Grpd$, are, of course, groupoid homomorphisms, and that groupoid homomorphisms also form (groupoid) [[../TrivialGroupoid/|functor categories]] defined in the standard manner for categories.