## Background

By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all $X \in C$. Recall that the internal hom $\underline{\mathrm{hom}}(X,-)$ is defined, if it exists, as a right adjoint of $- \otimes X$, and that $C$ is called closed if internal homs exist for all $X \in C$. According to the General Adjoint Functor Theorem, this reduces just to a size condition: For all $X,Y \in C$ there should be a *set* of morphisms $(Z_i \otimes X \to Y)_{i \in I}$, such that any other morphism $Z \otimes X \to Y$ factors as $Z \otimes X \to Z_i \otimes X \to Y$ for some $i \in I$ and $Z \to Z_i$.

## Questions

**Q1.** What is a nice example of a cocomplete symmetric monoidal category which is not closed?

A standard example for a cartesian category which is not closed is $\mathsf{Top}$; but $-\times \mathbb{Q}$ doesn't preserve coequalizers so that this doesn't answer the question (similar problems with other standard examples). If Q1 is too easy, what about cartesian categories?

**Q2.** What is a nice example of a cocomplete category with products, such that $-\times X$ is cocontinuous for all $X$, but has no right adjoint in general?

A weaker question would be:

**Q3.** What is a nice example of a cocomplete symmetric monoidal category whose underlying category is not locally presentable?

Again the standard examples of non-locally presentable categories which I have found in the literature don't fit here.

closedbicomplete symmetric monoidal categories that are not locally presentable. For instance, the category of compactly generated Hausdorff spaces. – Mike Shulman Jan 7 '13 at 8:03