How to define a left adjoint to a functor

Question

Let $\phi:H\to G$ be a group homomorphism. Then there exists a functor $\phi^*:Set^{BG}\to Set^{BH}$. I am struggling to define a left adjoint to this. May I get a hint?

Answer 1

Typically when trying to find a left adjoint to an algebraic construction, you want to "freely" add the required structure. In this case, $\phi^*$ takes $H$-sets (that is, sets with an $H$-action), and returns $G$-sets. It does this by defining $$g \cdot_G x := \phi(g) \cdot_H x$$

Now, given a $G$-set, we want to find a way to turn it into an $H$ set "freely". This means we want to disturb the structure of our $G$-set as little as possible. There is one obvious way to make any set $X$ into an $H$-set, and that is by turning $X$ into $H \times X$. Then we can define $h \cdot (h',x) := (hh',x)$. However, we want our set to continue playing nice with our $G$-set structure, and in this instance we cannot recover the $G$-set structure.

Experience (or experimentation) shows how we can solve this issue! We define $$\phi_!(X) = H \times X/(h \cdot \varphi(g),x) = (h,g \cdot x)$$ I will leave it to you to check that this actually satisfies the adjunction.

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I hope this helps ^_^