The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

Question

In 6.5 of the book by Kelly,

Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.

the author claims that the $2$-category $\mathsf{Cat}_{\mathcal{F}}$ of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors is in some sense monoidal closed. Indeed $\mathsf{Cat}_{\mathcal{F}}$ has a very natural notion of internal hom, because $\mathsf{Cat}_{\mathcal{F}}(A,B)$ is $\mathcal{F}$-cocomplete. Kelly's result shows that one can define a tensor $\otimes: \mathsf{Cat}_{\mathcal{F}} \times \mathsf{Cat}_{\mathcal{F}} \to \mathsf{Cat}_{\mathcal{F}}$ having the usual property of a monoidal closed structure (up to replacing isomorphisms with euquivalences of categories.

Q1: Unfortunately, I do not understand where $A \otimes B$ is defined, to my understanding he starts mentioning it, but I do not get where the definition is given. Could someone help me to understand it?

After a while, I think I got the definition (even if I do not find it in the book, and I wish someone can tell me what the author is doing there), which is quite involved and comes from a $2$-dimensional adaptation of a classical result in $1$-dimensional category theory. The result is mostly due to Kock and Seal, but I shall mention the Chap. 6 of PhD thesis of Martin Brandenburg, Tensor categorical foundations of algebraic geometry, because he gathered the existing literature in a coherent way.

Thm. (Seal, 6.5.1 in Ref.) Let $T$ be a coherent (symmetric) monoidal monad on a (symmetric) monoidal category C. Then $\mathsf{Mod}(T)$ becomes a (symmetric) monoidal category.

Seal does not show that it is monoidal closed, because he does not assume closedness of the base, yet he proves monoidality of the Eilenberg-More category of algebras. Hopefully, if the base is closed, so is the EM-category.

Q2: Unfortunately, to my understanding, the literature contains a plethora of notions of nice monads: coherent, commutative, monoidal, strong, Hopf... can someone guide me among these options? Which kind of monads induces a monoidal closed structure on the category of algebras?

Q3: I wanted to use a Kock-like result in the case in which $T$ is the free completion under $\mathcal{F}$-colimits over $\mathsf{Cat}$ in order to show that $\mathsf{Cat}_{\mathcal{F}} \cong T\text{-}\mathsf{Alg}$ is monoidal closed. Indeed to me, this is what Kelly is secretly doing. Does anyone know a reference that uses this kind of argument? Indeed here many $2$-dimensional and size-related subtleties should be taken into account.

Answer 1

Let $\mathcal{V}$ be a complete and cocomplete closed symmetric monoidal category. Let $\mathcal{F}$ be a small set of indexing types aka weights (these are just $\mathcal{V}$-functors from a $\mathcal{V}$-category to $\mathcal{V}$) and $\mathsf{cat}_{\mathcal{F}}$ denote the $2$-category of essentially small $\mathcal{V}$-categories which are $\mathcal{F}$-cocomplete; the following construction does not work for large categories. I denote the Hom-categories of $\mathcal{F}$-cocontinuous functors by $\mathrm{Hom}_{\mathcal{F}}$; Kelly writes $\mathcal{F}\mathrm{-Cocts}$.

If $\mathcal{C},\mathcal{D},\mathcal{E} \in \mathsf{cat}_{\mathcal{F}}$, let us call a $\mathcal{V}$-functor $\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$ bi-$\mathcal{F}$-cocontinuous if it preserves $\mathcal{F}$-colimits in each variable (see Section 1.4 in Kelly's book for the definition of $\otimes$ here). The tensor product $\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D}$ is defined by the universal property $$\mathrm{Hom}_{\mathcal{F}}(\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D},\mathcal{E}) \simeq \{\text{bi-}\mathcal{F}\text{-cocontinuous functors } \mathcal{C} \otimes \mathcal{D} \to \mathcal{E}\}.$$ Of course, these equivalences should be natural in $\mathcal{E}$. We can also write $$\mathrm{Hom}_{\mathcal{F}}(\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D},\mathcal{E}) \simeq \mathrm{Hom}_{\mathcal{F}}(\mathcal{C},\mathrm{Hom}_{\mathcal{F}}(\mathcal{D},\mathcal{E})).$$ The idea of the construction of this tensor product of categories is very similar to the construction of the tensor product of modules (take a free module on the product and then introduce the bilinear relations).

Here, we start with the free cocompletion $\widehat{\mathcal{C} \otimes \mathcal{D}}$ of $\mathcal{C} \otimes \mathcal{D}$, which is the $\mathcal{V}$-category of $ \mathcal{V}$-functors $(\mathcal{C} \otimes \mathcal{D})^{op} \to \mathcal{V}$. We have the Yoneda embedding $Y : \mathcal{C} \otimes \mathcal{D} \to \widehat{\mathcal{C} \otimes \mathcal{D}}$.

Let $\Phi$ denote the set of cylinders in $\mathcal{C} \otimes \mathcal{D}$ which are either of the form $$F \xrightarrow{\lambda} \mathrm{Hom}_\mathcal{C}(G(-),A) \xrightarrow{\text{ can }} \mathrm{Hom}_{\mathcal{C} \otimes \mathcal{D}}(G(-) \otimes B, A \otimes B)$$ for some indexing type $F : \mathcal{J}^{op} \to \mathcal{V}$ in $\mathcal{F}$, some $\mathcal{V}$-functor $G : \mathcal{J} \to \mathcal{C}$, some colimit cylinder $\lambda : F \to \mathrm{Hom}_\mathcal{C}(G(-),A)$ and some $B \in \mathcal{D}$, or of the form $$F \xrightarrow{\mu} \mathrm{Hom}_\mathcal{D}(H(-),B) \xrightarrow{\text{ can }} \mathrm{Hom}_{\mathcal{C} \otimes \mathcal{D}}(A \otimes H(-), A \otimes B)$$ for some indexing type $F : \mathcal{J}^{op} \to \mathcal{V}$ in $\mathcal{F}$, some $\mathcal{V}$-functor $H : \mathcal{J} \to \mathcal{D}$, some colimit cylinder $\mu: F \to \mathrm{Hom}_\mathcal{D}(H(-),B)$ and some $A \in \mathcal{C}$.

Now consider $\mathrm{Alg}(\Phi)$, the full subcategory of $\widehat{\mathcal{C} \otimes \mathcal{D}}$ which consists of those $\mathcal{V}$-functors $P : (\mathcal{C} \otimes \mathcal{D})^{op} \to \mathcal{V}$ which map all cylinders in $\Phi$ to limit cylinders in $\mathcal{V}$. It is a non-trivial result that this is a reflective subcategory (Theorem 6.5 in Kelly's book), in particular cocomplete. This also needs the extra assumption that $\mathcal{V}$ is locally bounded. Let $R : \widehat{\mathcal{C} \otimes \mathcal{D}} \to \mathrm{Alg}(\Phi)$ denote the reflector.

Define $\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D}$ as the smallest full subcategory of $\mathrm{Alg}(\Phi)$ which is closed under $\mathcal{F}$-colimits and contains the image of $$\mathcal{C} \otimes \mathcal{D} \xrightarrow{Y} \widehat{\mathcal{C} \otimes \mathcal{D}} \xrightarrow{R} \mathrm{Alg}(\Phi).$$ It is clear that $\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D}$ is an object of $\mathsf{cat}_{\mathcal{F}}$. It has the required universal property: Theorem 6.23 in Kelly's book says that $\mathrm{Hom}_{\mathcal{F}}(\mathcal{C} \boxtimes_{\mathcal{F}} \mathcal{D},\mathcal{E})$ is equivalent to the category of $\Phi$-comodels in $\mathcal{E}$, which by definition are $\mathcal{V}$-functors $\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$ which map the cylinders in $\Phi$ to colimit cylinders. According to the definition of $\Phi$, these are exactly the bi-$\mathcal{F}$-cocontinuous functors $\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$.

Answer 2

I am not aware of Kock's works.

Nevertheless Kelly provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\mathcal F$-theory generated by the sketch $((\mathcal A \otimes \mathcal B)^{op},\Phi)$ where $\Phi$ is made of the $\mathcal A \otimes \mathcal B$ cylinder of the form $\lambda \otimes B$ and $A \otimes \mu$ defined in the previous paragraph ($\lambda$ and $\mu$ range over the family of colimit cylinders of $\mathcal A$ and $\mathcal B$ respectively).

Edit: I see your problem was not with the tensor in $\mathcal F-\mathbf{Cat}$ the tensor $\mathcal A \otimes \mathcal B$. This is the tensor product of $\mathcal A$ and $\mathcal B$ as $\mathcal V$-categories, the definition can be found in section 1.4 page 12.

I think it is important to stress the fact $\mathcal A \otimes \mathcal B$ is not a $\mathcal F$-complete category.

I hope this helps.