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

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.

• Could you give a reference for Kock's results ? he has several paper on strong & commutative monads and I can't find the one where this theorem is... – Simon Henry Jan 21 at 16:28
• @SimonHenry, thanks. I edited. – Ivan Di Liberti Jan 21 at 16:49
• I think the reference you gave only show that the category of algebra is a "closed category", it does not show it is monoidal. I will be tempted to think that to get a monoidal category of algebras you need some presentabitliy/accessibility assumption, or at least cocompletness. In the case of interest to you this result only gave you that functor categories are $\mathcal{F}$-cocomplete, which you already know. – Simon Henry Jan 21 at 17:02
• @SimonHenry, I hope I finally managed to give a good reference. 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. No assumption on presentability is needed. I believe that if the base is closed, so is the EM-category. – Ivan Di Liberti Jan 21 at 17:31
• @IvanDiLiberti have you seen Bourke's paper on Skew structures in 2-category theory and homotopy theory? Section 6 discusses a 2-dimensional version of Kock's result due to Hyland and Power, which Bourke argues is nicely analyzed via a skew closed structure on the 2-category $T\text{-}\mathrm{Alg}_s$ of algebras and strict morphisms. – Noam Zeilberger Jan 21 at 21:04

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 $$\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$$.

• Just a doubt. Should $\mathcal C \boxtimes_{\mathcal F}\mathcal D$ be a subcategory of $\text{Alg}(\Phi)$, rather than being a subcategory of $\widehat {\mathcal C \otimes \mathcal D}$? – Giorgio Mossa Jan 25 at 0:09
• @GiorgioMossa Yes, thanks! It was a typo. – Martin Brandenburg Jan 25 at 0:16

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.

• This is more or less a copy-paste of Kelly's words, but I frankly do not get them. $A \otimes B$ should be a $V$-category with $\mathcal{F}$-colimits, not a theory, nor a sketch. I am probably missing something incredibly simple, but I do not understand Kelly's words. – Ivan Di Liberti Jan 21 at 19:20
• I've made an edit. It should address your doubts. – Giorgio Mossa Jan 21 at 19:47
• I think I understood, but I will not accept this answer, hopefully will come with a more explanatory answer. This answers Q1, but does not make Kelly more readable at all. That was the main point of my question. – Ivan Di Liberti Jan 21 at 20:29
• I think it is not fair to wait for an answer which answers all three questions. I suggest to open a new thread for the whole monad thing (which is, in fact, not necessary for Kelly's tensor product). – Martin Brandenburg Jan 23 at 10:48