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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.

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.

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 we 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.

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.

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.

Thm. (Kock) Let $T$ be a commutative monad over a symmetric monoidal closed category $C$, then its category of algebras is symmetric monoidal closed.

This appears as Thm. 2.2 in Kock, Closed categories generated by commutative monads, Journal of the Australian Mathematical Society. Volume 12, Issue 4 November 1971, pp. 405-424.

Another relevant result, polishing the exposition in the previous paper is Thm. 4 in Kock, Commutative monads as a theory of distributions, TAC, 26 (4) (2012), pp 97-131.

Q2: Unfortunately, to my understanding, the literature contains a plethora of notions of nice monads: 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.

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 we 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.