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I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras of commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of limit theories, including $\rm{Cat}$ theory, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras over a commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of limit theories, including $\rm{Cat}$ theory, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras of commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of locally presentable categories, including $\rm{Cat}$, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras of commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of limit theories, including $\rm{Cat}$ theory, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras of commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$. Is there any general reason why $\mathrm{Cat}$ has a (Cartesian!) symmetric closed monoidal structure? And $\mathrm{Pos}$ (category of posets)?

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
2. Category algebras of commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of locally presentable categories, including $\rm{Cat}$, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?