TITLE: Left G-Sets category
QUESTION [6 upvotes]: Let $G$ be a group, and $\mathbf{G\text{-}Sets}$ the category whose objects are left G-Sets and whose morphisms are G-Set homomorphisms, that is functions $f:X\to Y$ such that $f(ax) = af(x)$, $a\in G, x\in X$.
Does the forgetful functor $\mathbf{G\text{-}Sets}$ $\to$ $\mathbf{Sets}$ have a left-adjoint?
Are there any other interesting functors between the categories $\mathbf{G\text{-}Sets}$ and $\mathbf{Sets}$?

REPLY [3 votes]: If $H\le G$ is a subgroup let $\mathrm{Res}_H^G:G\textrm{-}\mathsf{Set}\to H\textrm{-}\mathsf{Set}$ be the restriction functor. One can think of $\mathsf{Set}$ as the case where $H=1$ is the trivial subgroup. It has a left adjoint, call it $\mathrm{Ind}_H^G$, so that
$$\hom_{G\textrm{-}\mathsf{Set}}(\mathrm{Ind}_H^G-,-) ~~\cong~ \hom_{H\textrm{-}\mathsf{Set}}(-,\mathrm{Res}_H^G-)$$
are naturally isomorphic functors $H\textrm{-}\mathsf{Set}\times G\textrm{-}\mathsf{Set}\to\mathsf{Set}$.
An explicit construction is $\mathrm{Ind}_H^G\Omega\cong G\times_H \Omega$, which is $G\times\Omega$ (the action of $G$ purely on the first coordinate) modulo the relation $(gh,\omega)\sim(g,h\omega)$ for all $g\in G,h\in H,\omega\in\Omega$.
Does this look familiar? If you've studied representation theory, it should. The linearized version of this fact is called Frobeius reciprocity. Let $G\textrm{-}\mathsf{Rep}$ be the category of linear representations of $G$, and then define the restriction functor $\mathrm{Res}_H^G:G\textrm{-}\mathsf{Rep}\to H\textrm{-}\mathsf{Rep}$. Frobenius reciprocity (the categorical version of it, as opposed to the more standard, decategorified, character-theoretic version of it) states that restriction has a left adjoint called induction $\mathrm{Ind}_H^G$, so $\hom_G(\mathrm{Ind}_H^GV,W)\cong\hom_H(V,\mathrm{Res}_H^GW)$ canonically for all representations $V$ of $G$ and $W$ of $H$.
An explicit construction is $\mathrm{Ind}_H^GV\cong \Bbb C[G]\otimes_{\Bbb C[H]}V$, the linearized analogue of $G\times_H\Omega$!