Adjoints between the category of sets and the category of left G-sets.

Question

What functors are there between the category Set and the category of left G sets? Of these functors can you show which are adjoints?

Bonus points for leading this question, and me, in a more rigorous direction.

Answer 1

The forgetful functor from $G$-sets to sets has adjoints on both sides. The left adjoint sends any set $A$ to the product $G\times A$ with the $G$-action defined by $g(h,a)=(gh,a)$ (for all $g,h\in G$ and $a\in A$). This should be called the free $G$-set generated by $A$. The right adjoint sends $A$ to the set of all functions from $G$ to $A$ (just set-theoretic functions, ignoring the group structure of $G$) with the $G$-action defined by $(gf)(h)=f(hg)$ for all $g,h\in G$ and all functions $f:G\to A$.

Answer 2

Going from G-sets to sets represents a loss of structure. This is the nature behind the forgetful functor designation; it "forgets" what it means to be "G" and only remembers what it means to be "-set".

And in an example of duality, in G-sets this functor has an opposite. And while the conventional way of thinking about opposites is in terms of a forward and backward, positive and negative, I suggest to you that in category theory it's more fruitful to think of these adjoints as "doing" and "undoing"

As such, if in one direction this adjunction represents a loss of structure (the doing) then going in the other direction must represent a gain in structure (undoing that loss).

The gain in structure that compliments a forgetful loss is known as the FREE FUNCTOR.

The idea behind the free functor is that in going in the opposite direction, from sets to G-sets, you only add the minimal amount of structure to uniquely identify a G-set from any other set in your theory. This is known as a free construction since it is "free" of any superfluous structure except for "G".

(PS: As a comment made clear, this is NOT true of all categories, i.e., not all categories with a forgetful functor have a free functor)

Answer 3

The relevant functors out of $G$-sets are the fixed point functor, which has been mentioned in comments, and the orbit functor $X\mapsto X/G$, which has not. One can generalize these to the fixed points or orbits with respect to any subgroup. The orbit functor is left adjoint to the free $G$-set functor, while fixed points are right adjoint to the trivial $G$-set functor. This becomes less muddled when one observes that the forgetful functor is simply fixed points for the trivial subgroup, and that the trivial $G$-set on $S$ is (essentially) just the free trivial group action on $S$.