Precomposition in Functor Categories: Restriction of Group Actions and Restriction of Scalars

Let $[\mathcal{C},\mathcal{D}]$$[\mathcal C,\mathcal D]$ be a functor category, we could consider $[-,\mathcal{D}]:={\mathrm{Hom}}_{\mathrm{Cat}}(-,\mathcal{D})$$[-,\mathcal D]:=\mathrm{Hom}_{\mathrm{Cat}}(-,\mathcal D)$

Let $\mathcal{F}:{\mathcal{C}}^{\prime }\to \mathcal{C}$$\mathcal F:\mathcal C'\to\mathcal C$ be a functor, then we get ${\mathcal{F}}^{\ast }:[\mathcal{C},\mathcal{D}]\to [{\mathcal{C}}^{\prime },\mathcal{D}],{\mathcal{F}}^{\ast }(G)=G\circ F$$\mathcal F^*:[\mathcal C,\mathcal D]\to [\mathcal C',\mathcal D],\mathcal F^*(G)=G\circ F$.

Recall that in Category of G-Set and Category of R-Module: A Uniform Approach via Category Theory

We show that category of $G$$G$-Object and $R$$R$-Module are functor category $[B(G),\mathcal{C}]$$[B(G),\mathcal C]$ and $[B(R):\mathrm{Ab}]$$[B(R):\mathrm{Ab}]$.

Then we could apply it to these particular examples. If you apply it on $[B(R):\mathrm{Ab}]$$[B(R):\mathrm{Ab}]$, then you get the restriction of scalar functor.

 

The Orbit Functor and Its Right Adjoint

Let us consider a functor $\mathcal{O}:G$$\mathcal O:G$-$\mathbf{Set}\to \mathbf{Set}$$\mathbf{Set}\to\mathbf{Set}$.

For a $G$$G$-Set $X$$X$, $\mathcal{O}(X):=X/G$$\mathcal O(X):=X/G$ is the set of orbits. For an $G$$G$-map $f$$f$, we have $f(gx)=gf(x)$$f(gx)=gf(x)$ hence it maps orbit of $x$$x$ to orbit of $f(x)$$f(x)$.

This functor admits a right adjoint, $\mathrm{\Delta }:\mathbf{Set}\to$$\Delta:\mathbf{Set}\to$$G$$G$-$\mathbf{Set}$$\mathbf{Set}$. For each set $X$$X$, $\mathrm{\Delta }(X)$$\Delta(X)$ is the $G$$G$-Set with trivial action.

$${\mathrm{Hom}}_{\mathbf{Set}}(\mathcal{O}(X),Y)\cong {\mathrm{Hom}}_{G-\mathbf{Set}}(X,\mathrm{\Delta }(Y))$$

You could replace $\mathbf{Set}$$\mathbf{Set}$ by $\mathbf{Top}$$\mathbf{Top}$ to get similar results.