In **6.5** of the book by *Kelly*,



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



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> **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?
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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*,](https://arxiv.org/abs/1410.1716) **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.



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> **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**?
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> **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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The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors
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 $\mathcal{C} \otimes \mathcal{D} \to \mathcal{E}$.


