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.



Example of a non-closed cocomplete symmetric monoidal category
Here is an amusing example which addresses Q2: take the universe $V$ of sets in a model of ZFC, as a class partially ordered by inclusion of sets. Consider a partially ordered class to be a category in the usual way. Of course, by Cantor's theorem, there is no terminal object in this category, but anyway we can freely adjoin one; let $V\_+$ denote the result. Notice that cartesian products are given by taking intersections. 


The category $V\_+$ is small cocomplete since we can take small unions, and of course intersections $- \cap X$ distribute over unions, so we get a cocomplete cartesian monoidal category. But I claim that unless $X$ is the top element (that we freely adjoined), there is no exponential $Y^X$ for any strict subset $Y \subset X$. Indeed, if $Z = Y^X$, then $Z$ would have to be the largest set such that $Z \cap X \subseteq Y$. But there is no such largest set, since to any such set we could add more elements which do not belong to $X$ to get a larger set $Z'$, and we'd still have $Z' \cap X \subseteq Y$! 


