Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits?

Question

Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\mathrm{Topos}$ which assigns $M$ to the category of sets with $M$-action ($M\text{-Sets}$). Does this preserve limits?

Answer 1

The answer is no, but there is a subtlety here.

Since the category of commutative monoids is closed under limits in the category of monoids, and because it is annoying to have to write "commutative" everywhere, I will just talk about monoids. As a warm-up, consider the functor $\mathbb{B} : \textbf{Mon} \to \textbf{Cat}$ sending each monoid to the 1-object category it corresponds to. It is easy to check that $\mathbb{B}$ preserves limits ... if by $\textbf{Cat}$ we mean the 1-category of categories. But if $\textbf{Cat}$ means the 2-category of categories, and by "limit" in $\textbf{Cat}$ we mean the bicategorical notion, then $\mathbb{B}$ does not even preserve pullbacks. Indeed, for any monoid $M$, $$\require{AMScd} \begin{CD} \coprod_{M_\textrm{inv}} \mathbb{B} 1 @>>> \mathbb{B} 1 \\ @VVV \Rightarrow @VVV \\ \mathbb{B} 1 @>>> \mathbb{B} M \end{CD}$$ is a bipullback diagram, where $M_\textrm{inv}$ is the set of invertible elements of $M$ and the isomorphism of composites $\coprod_{M_\textrm{inv}} \mathbb{B} 1 \to \mathbb{B} M$ assigns to each element of $M_\textrm{inv}$ the corresponding morphism in $\mathbb{B} M$.

A similar phenomenon is seen for the pseudofunctor $\mathbf{B} : \textbf{Mon} \to \textbf{Topos}$ sending a monoid $M$ to the topos $\mathbf{B} M$ of right $M$-sets, but the calculations are more complicated. Let $M_M$ be $M$ considered a right $M$-set. Note that $M_M$ has a left $M$-action, considered as a right $M$-set. $\mathbf{B} M$ is freely generated under colimits by $M_M$, so in particular, a geometric morphism $f : \mathcal{E} \to \mathbf{B} M$ is uniquely determined (up to isomorphism) by $f^* M_M$, which is an object in $\mathcal{E}$ with a left $M$-action. Diaconescu's theorem tells us which such objects give rise to geometric morphisms. In the case $\mathcal{E} = \mathbf{B} 1$, a left $M$-set $T$ induces a geometric morphism if it is flat, i.e.

• $T$ has at least one element, and
• Given any $t_0$ and $t_1$ in $T$, there exist $m_0$ and $m_1$ in $M$ such that $m_0 \cdot t_0 = m_1 \cdot t_1$, and
• Given any $t$ in $T$ and any $m_0$ and $m_1$ in $M$ such that $m_0 \cdot t = m_1 \cdot t$, there exists $m_2$ in $M$ such that $m_2 m_0 = m_2 m_1$.

If $G$ is a group then a flat left $G$-set is precisely a left $G$-set with a free transitive action. There is only one such thing up to isomorphism, namely ${}_G G$, so there is only one geometric morphism $\mathbf{B} 1 \to \mathbf{B} G$ up to isomorphism. However, the isomorphism is not unique! Indeed, each element of $G$ gives rise to a different isomorphism ${}_G G \to {}_G G$. On the other hand, $\mathbf{B} 1$ is terminal, so we conclude that $$\begin{CD} \coprod_{G} \mathbf{B} 1 @>>> \mathbf{B} 1 \\ @VVV \Rightarrow @VVV \\ \mathbf{B}1 @>>> \mathbf{B} G \end{CD}$$ is a bipullback diagram. Concretely, $\coprod_{G} \mathbf{B} 1$ is the topos of $G$-indexed families of sets, the composites $\coprod_{G} \mathbf{B} 1 \to \mathbf{B} G$ have inverse image functor given by sending a right $G$-set $X$ to $(X, X, X, \ldots)$, and the isomorphism twists the $g$-th component by $g$ (on the right, of course).