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 $\mathbf{Sets}$?
Best Answer
Let $S$ be a set. Then $G \times S$ can be equipped with a $G$-action by having $G$ act by left-multiplication on itself and trivially on $S$. This is a left-adjoint to the forgetful functor -- for $T$ an arbitrary $G$-set, one has
$$ \text{Hom}_\text{G-set}(G \times S, T) = \text{Hom}_\text{set}(S, T) $$ by the rule $f \mapsto \left[s \mapsto f(1, s)\right]$. The map the other way is $\phi \mapsto \left[(g, s) \mapsto g\phi(s)\right]$ .