CatGloss

The objects of the functor category ,of, of G$-representations and equivariant linear maps}$\textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G}are are Grepresentationsoverthefield-representations over the field \mathbbe{k},andarrowsare, and arrows are Gequivariantlinearmaps.If-equivariant linear maps. If Hisasubgroupof is a subgroup of G,restriction, restriction \textup{fun}_H^G \colon \textup{\textsf{cat}}\mathbbe{k}^{\mathsf{B} G} \to \textup{\textsf{cat}}_\mathbbe{k}^{\mathsf{B} H}ofa of a Grepresentationtoan-representation to an Hrepresentationissimplyprecompositionbytheinclusionfunctor-representation is simply pre-composition by the inclusion functor \mathsf{B} H \hookrightarrow \mathsf{B} G$. This functor has a left adjoint, called induction, and a right adjoint, called coinduction. \(\vcenter{\xymatrix{\cat{Vect}_\mathbbe{k}^{\mathsf{B} G} \ar[r]\mid{\mathrm{res}_H^G} & \textup{\textsf{cat}}_\mathbbe{k}^{\mathsf{B} H} \ar@/^1.5pc/[l]^{\mathrm{coind}_H^G} \ar@/_1.5pc/[l]_{\mathrm{ind}_H^G} \ar@{}[l]^*+{\labelstyle{\perp}}_*+{\labelstyle\perp} }}\) By Proposition \ref{prop:kan-adjoint}, the induction functor is given by left Kan extension along the inclusion BHBG\mathsf{B} H \hookrightarrow \mathsf{B} G, while coinduction is given by right Kan extension. The reader unfamiliar with the construction of induced representations need not remain in suspense for very long; see Theorem \ref{thm:Kan-formula} and Example \ref{ex:repKan2}. Similar remarks apply for GG-sets, GG-spaces, based GG-spaces, or indeed GG-objects in any category, although if the ambient category has few limits and colimits, these adjoints need not exist.