Editing Groupoid (section)

=== Relation to [[Category of small categories|Cat]] ===

The inclusion <math>i : \mathbf{Grpd} \to \mathbf{Cat}</math> has both a left and a right [[Adjoint functors|adjoint]]:
: <math> \hom_{\mathbf{Grpd}}(C[C^{-1}], G) \cong \hom_{\mathbf{Cat}}(C, i(G)) </math>
: <math> \hom_{\mathbf{Cat}}(i(G), C) \cong \hom_{\mathbf{Grpd}}(G, \mathrm{Core}(C)) </math>

Here, <math>C[C^{-1}]</math> denotes the [[localization of a category]] that inverts every morphism, and <math>\mathrm{Core}(C)</math> denotes the subcategory of all isomorphisms.