# Adjoints for the restriction of category-valued representations of groups

Setup. Let $$G$$ be a group and let $$\mathscr{A}$$ be a category. We denote the category of functors from $$G$$ to $$\mathscr{A}$$ by $$[G, \mathscr{A}]$$ and think of these functors as $$\mathscr{A}$$-valued representation of $$G$$. (More explicitely, such a representation consists of an object $$X$$ of $$\mathscr{A}$$ and a homomorphism of groups from $$G$$ to $$\operatorname{Aut}_{\mathscr{A}}(X)$$.)

Let now $$H$$ be a subgroup of $$G$$. The inclusion map $$i \colon H \to G$$ can be regarded as a functor, which then induces a functor $$\operatorname{res}^G_H ≔ i^* \colon [G, \mathscr{A}] \longrightarrow [H, \mathscr{A}] \,.$$ This functor restricts the $$\mathscr{A}$$-valued representations of $$G$$ to $$\mathscr{A}$$-valued representations of $$H$$.

Question. Under what conditions (on $$\mathscr{A}$$) does the restriction functor $$\operatorname{res}^G_H$$ admit a left adjoint, resp. a right adjoint?

This question is motivated by Exercise 2.1.16 in Tom Leinster’s Basic Category Theory, which deals with $$[G, \mathrm{Set}]$$ and $$[G, \operatorname{Vect}(\mathbb{k})]$$.

I believe that I understand the following two special cases:

• For every ring $$R$$ we have for $$\mathscr{A} = \operatorname{Mod}(R)$$ that $$[G, \mathscr{A}] ≅ \operatorname{Mod}(R[G])$$. We thus have the usual adjunctions $$R[G] \otimes_{R[H]} (-) ⊣ \operatorname{res}^G_H ⊣ \operatorname{Hom}_{R[H]}(R[G], -) \,.$$
• In the case of $$\mathscr{A} = \mathrm{Set}$$ we have similarly $$[G, \mathscr{A}] ≅ G\textrm{-}\mathrm{Set}$$ and adjunctions $$G \times_H (-) ⊣ \operatorname{res}^G_H ⊣ \operatorname{Hom}_H(G, -) \,.$$

However, I don’t expect these examples to generalize to an arbitrary category $$\mathscr{A}$$, as they both rely on some notion of tensor-hom adjuction.

• Well, existence of limits/colimits is enough - the adjoints will be Kan extensions. Oct 31, 2021 at 2:49
• These are subsumed by the notion of Kan extension, for which there are satisfactory existence theorems. I’m sure they’re covered in the book, though maybe later. Oct 31, 2021 at 2:50
• About Kan extensions: In Leinster‘s book Kan extensions are only briefly mentioned in Exercise 6.2.25. Oct 31, 2021 at 4:09
• Example 6.2.8 in Emily Riehl's "Category Theory in Context" discusses this example btw. Oct 31, 2021 at 8:47

This is explained in Emily Riehl’s book “Category Theory in Context”, in particular in Example 6.2.8, as mentioned in the comments. More explicitely, a left adjoint exists if $$\mathscr{A}$$ is cocomplete, and a right adjoint exists if $$\mathscr{A}$$ is complete.