PlanetPhysics/Groupoid Categories

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.