Presentability rank of tensor product of presentable categories

Question

In this post category means $(\infty, 1)$-category.

Let $X, Y$ be two presentable categories. We can then form their tensor product $X \otimes Y \cong \operatorname{ContFun}(X^{\mathrm{op}}, Y)$. Can we bound the presentability rank of $X \otimes Y$ in terms of that of $X, Y$? I'm particularly interested in whether the tensor product of two finitely presentable categories is still finitely presentable. If we write $X \cong \operatorname{Ind}^\omega(T^{\mathrm{op}})$ for a finitely complete small category $T$ then $X \otimes Y \cong \operatorname{Lex}(T, Y) = T-\mathrm{Mod}(Y)$, so I guess my question is equivalent to whether the category of models of a finite limit theory in a finitely presentable category is still finitely presentable.

The nlab page for the tensor product of presentable categories says $\operatorname{Lex}(T, Y)$ is an accessible localization of $\operatorname{Fun}(T, Y)$, so if the localization functor is finitary we'd get the result we want. But I don't actually know why this is acessibly embedded (the proof of presentability in Higher Algebra uses a different method).

Answer 1

The answer is yes, you can bound it by the maximum of the two accessibility ranks.

The point is the following: let $\otimes_\kappa$ denote the Lurie tensor product on $Cat_\kappa$, the category of categories with $\kappa$-small colimits and $\kappa$-cocontinuous functors between them. This admits a tensor product classifying functors that are $\kappa$-cocontinuous functors in each variable for general reasons.

So now let $X,Y$ be $\kappa$-compactly generated. We have $X= Ind_\kappa(X^\kappa), Y= Ind_\kappa(Y^\kappa)$, and I want to prove that $X\otimes Y = Ind_\kappa(X^\kappa \otimes_\kappa Y^\kappa)$ - for this, I will prove that they have the same universal property in $Pr^L$ : $$Fun^L(X\otimes Y, Z) \simeq Fun^L(X,Fun^L(Y,Z)) \simeq Fun^L(X,Fun^{<\kappa}(Y^\kappa, Z)) \simeq Fun^{<\kappa}(X^\kappa, Fun^{<\kappa}(Y^\kappa,Z))\simeq Fun^{<\kappa,<\kappa}(X^\kappa \times Y^\kappa, Z) \simeq Fun^{<\kappa}(X^\kappa\otimes_\kappa Y^\kappa, Z)\simeq Fun^L(Ind_\kappa(X^\kappa\otimes_\kappa Y^\kappa), Z)$$

(if you're worried about the second-to-last equivalence because $Z$ is not small, note that you can simply filter by the $Z^\lambda$'s, $\lambda\geq \kappa$)

It is a bit surprising: I can't show directly that if $x\in X^\kappa, y\in Y^\kappa$, then $x\otimes y\in (X\otimes Y)^\kappa$ but it follows from this proof.