# Is there a "duality involution" on presentable categories?


Question: Is there an involution $$(-)^\star : \Pr^L \to \Pr^L$$ such that $$i(\Psh(C)^\ast) = (i(\Psh(C)))^\star$$?

(I have freely mixed and matched terminology here from 1-categories and $$\infty$$-categories. The above question is really two questions: one in the 1-categorical case and another in the $$\infty$$-categorical case. Please ask if it's unclear what I'm saying!)

Notes:

• Of course, the opposite category to a presentable category is rarely presentable. Note that the putative involution I'm asking about would not be obtained by taking the opposite category.

• The duality involution on $$\Psh$$ is related to dualizability with respect to the Lurie tensor product. I'm pretty sure I've been told that the only dualizable objects in $$\Pr^L$$ are the retracts of presheaf categories. But I don't think that rules out an involution of the form I'm asking about. (I'm a bit confused on this point too, because the involution I'm asking about would anyway be covariant rather than contravariant like the one related to dualizability.)

• Isn't $Psh$ equivalent to $\mathbf{Cat}$ under the requirement that functors preserve tiny objects? May 3 at 13:38
• @varkor Yes, $Psh$ is equivalent to the category of small, idempotent-complete categories and all functors. May 3 at 13:38
• In the 1-categorical case, I think this is the "reflexivity" question posed here by Brandenburg, Chirvasitu, and Johnson-Freyd? They show there's a pretty broad class of categories where you get reflexivity even when you don't have dualizability. They don't give any counterexamples that aren't reflexive as far as I can tell. May 3 at 16:09
• Oh, key point I missed, they're working in the k-linear setting which it looks like you aren't. May 3 at 16:55
• @NoahSnyder : there's also the question of variance: taking duals is contravariant (although Tim already pointed this out). May 3 at 18:52

The answer is no, even if you restrict to the full subcategory of $$Pr^L$$ spanned by the $$Psh(C)$$'s. I'll answer in the $$1$$-categorical case but : a- the $$\infty$$-categorical case follows because presentable $$1$$-categories are presentable $$\infty$$-categories and b- even if it didn't strictly follow, one easily convinces oneself that the same method works.

Indeed, your involution provides, for any $$Set\to Psh(C)$$, a functor $$Set = i(Set) \to i(Psh(C)) = Psh(C^{op})$$, i.e., for any presheaf $$F$$ on $$C$$, a canonical presheaf on $$C^{op}$$.

Specifically, for every $$F: C^{op} \to Set$$ it gives you some $$\iota F : C\to Set$$ in a way compatible with left Kan extension along small functors $$C\to D$$. Note that on representables, it sends $$\hom(-,x)$$ to $$\hom(x,-)$$.

Now I claim that $$\iota$$ can be extended to a functor.

Namely, say I have a natural transformation of presheaves of $$C$$, $$F\to G$$, viewed as $$\Delta^1 \to Psh(C)$$, then I can extend it to $$Psh(\Delta^1) \to Psh(C)$$ and the two inclusions $$\Delta^{\{i\}}\to \Delta^1$$ show that applying my involution $$i$$ and restricting along $$\Delta^1\to Psh(\Delta^1)$$ gives me a transformation $$\iota G\to \iota F$$ (there is an inversion of direction because of $$\Delta^1$$ vs $$(\Delta^1)^{op}$$.

Furthermore, by looking at $$\Delta^2$$, it is easy to see that this really makes $$\iota$$ into a functor. In particular $$\iota : Psh(C)\to Psh(C^{op})^{op}$$ is a functor which restricts to the identity along the Yoneda embeddings.

Because $$i$$ is an involution and not only a functor, you can do the same thing in the opposite direction, and the fact that it's an involution shows that the composite $$Psh(C)\to Psh(C^{op})^{op} \to Psh(C)$$ is the identity, and same of course in the other direction. In particular, $$Psh(C)\simeq Psh(C^{op})^{op}$$, which is impossible.


Is there a duality involution on $$\mathbf{Rex}$$ such that the following square commutes up to pseudonatural equivalence (where the 2-functor $$\mathbf{Cat} \to \mathbf{Rex}$$ is the free cocompletion under finite colimits)?

$$\require{AMScd}\begin{CD} \mathbf{Rex}^{\text{co}} @>{(-)^{\star}}>> \mathbf{Rex} \\ @AAA @AAA \\ \mathbf{Cat}^{\text{co}} @>>{(-)^{\text{op}}}> \mathbf{Cat}. \end{CD}$$

In this form, it seems unlikely there will be such a duality involution (certainly taking the opposite category does not work). However, if instead of considering the finitely cocomplete categories, we consider those categories with finite colimits and finite limits, a similar square does commute. By applying Gabriel–Ulmer duality again, we should get a duality involution on the sub-2-category of $$\Pres^L$$ whose objects have finitely complete subcategories of finitely presentable objects, and whose functors preserve finite limits, which satisfies the (pseudo)commutativity condition you are looking for.