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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.

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  • $\begingroup$ 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... $\endgroup$ – Simon Henry Jan 21 at 16:28
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    $\begingroup$ @SimonHenry, thanks. I edited. $\endgroup$ – Ivan Di Liberti Jan 21 at 16:49
  • $\begingroup$ 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. $\endgroup$ – Simon Henry Jan 21 at 17:02
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    $\begingroup$ @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. $\endgroup$ – Ivan Di Liberti Jan 21 at 17:31
  • $\begingroup$ @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. $\endgroup$ – Noam Zeilberger Jan 21 at 21:04
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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 $\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$.

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

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  • $\begingroup$ 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. $\endgroup$ – Ivan Di Liberti Jan 21 at 19:20
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    $\begingroup$ I've made an edit. It should address your doubts. $\endgroup$ – Giorgio Mossa Jan 21 at 19:47
  • $\begingroup$ 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. $\endgroup$ – Ivan Di Liberti Jan 21 at 20:29
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    $\begingroup$ 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). $\endgroup$ – Martin Brandenburg Jan 23 at 10:48

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