The title is a little bit imprecise. Consider "typical" categories $\mathcal{A},\mathcal{B}$, let's say infinite and locally small. A pair of adjointable functors $(F,G$) is a pair of functors $(F : \mathcal{A} \to \mathcal{B}, G : \mathcal{B} \to \mathcal{A})$, such that $F$ is left-adjoint to $G$. We consider $(F,G)$ and $(F',G'$) the same, if $F\cong F'$ or $G\cong G'$.

How many pairs of adjointable functors are there between $\mathcal{A}$ and $\mathcal{B}$?

I mean this very broadly: finitely many, infinitely many, possibly a proper class? If one cannot say anything generally, then what about important examples like $\mathcal{A},\mathcal{B}$ being categories of universal algebras or one being the category of small categories?


Let $A, B$ both be $\text{Set}$. For any set $X$ there is an adjunction between the functors $(-) \times X$ and $[X, -]$ (here I mean the set of functions from $X$ into some other set). So in this case there is a proper class of adjunctions, even up to isomorphism. I expect that this is typical. (Note that $F \cong F'$ is equivalent to $G \cong G'$.)

More generally in this argument we can replace $A, B$ with a closed monoidal category which is not essentially small. There are many examples, such as $(\text{Ab}, \otimes)$ or, for that matter, $(\text{Cat}, \times)$.

Edit: It's sometimes possible to classify all left adjoints $F : A \to B$. For example, if $A$ is the category $[C^{op}, \text{Set}]$ of presheaves on an essentially small category $C$, then the category of left adjoints $A \to B$ is equivalent to the category of functors $C \to B$. If $B$ itself is the category $[D^{op}, \text{Set}]$ of presheaves on another essentially small category $D$, then this is in turn the category $[C \times D^{op}, \text{Set}]$ of bimodules over $C$ and $D$.

  • $\begingroup$ Ah yes, I actually knew this example, but I completely missed that. $\endgroup$ – Stefan Perko Jan 12 '16 at 16:31

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