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What can we say about a functor that has both left and right adjoints?

I vaguely recall hearing that it is then an equivalence of category. Is it true?
If not, then under what conditions it is true?

Najib Idrissi
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mez
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1 Answers1

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No, it's not necessarily an equivalence of categories. For example, if $\varphi: R\to S$ is a homomorphism of rings, the forgetful functor $S\text{-Mod}\to R\text{-Mod}$ always has both the left adjoint $\text{Ind}_R^ S := -\otimes_R S$ and the right adjoint $\text{CoInd}_R^S:=\text{Hom}_R(S,-)$. It can even happen that these coincide without $S\text{-Mod}\to R\text{-Mod}$ being an equivalence: for example, if $R={\mathbb k}$ is a field and $S = {\mathbb k}G$ is the group algebra of a finite group $G$ over ${\mathbb k}$, then $\text{Ind}_{\mathbb k}^{{\mathbb k}G}\cong\text{CoInd}_{\mathbb k}^{{\mathbb k}G}$, but still ${\mathbb k}G\text{-Mod}\to{\mathbb k}\text{-Mod}$ is not an equivalence if $|G|>1$.

It also happens often that the inclusion of a subcategory has both a left and a right adjoint.

I don't know of a useful criterion telling you that a functor which has both left and right adjoints + satisfies certain extra properties has to be an equivalence of categories.

Hanno
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  • Aside from the question, can you give me some examples where the inclusion of a subcategory has both left and right adjoints? – mez Jan 07 '15 at 13:20
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    @mez: For example, if $\varphi:R\to S$ is surjective then $S\text{-Mod}\to R\text{-Mod}$ is fully faithful. – Hanno Jan 07 '15 at 13:27
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    For another example, consider the forgetful functor $\mathsf{Top} \to \mathsf{Set}$ sending a topological space to its underlying set. The left adjoint equips a set with the discrete topology while the right adjoint equips a set with the indiscrete topology. Similar "discrete/indiscrete" adjoints also occur for the forgetful functors $\mathsf{Cat} \to \mathsf{Set}$, $\mathsf{Pos} \to \mathsf{Set}$, $\mathsf{Grpd} \to \mathsf{Set}$. – tcamps Jan 07 '15 at 19:43
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    I've just learned from this nice webpage another interesting example: the functor between poset categories $\Bbb Z \to \Bbb R$ (incuced by the natural inclusion) has both a left and a right adjoint, resp. ceiling and floor functions! – Watson Aug 27 '18 at 12:39
  • Another example: if $C$ is the category of commutative rings of characteristic $p$ (for some fixed prime $p$) and $P$ is the subcategory consisting of perfect rings, then the forgetful functor $U : P \to C$ has a left adjoint (perfect closure) and a right adjoint (perfection), as mentioned here. – Watson Sep 24 '18 at 15:10