Simplicially enriched cartesian closed categories

Question

In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ordinary category is cartesian closed, but such that $C$ is not $V$-cartesian closed. The answers gave a general way to find such examples: let $P$ be a small cartesian monoidal category and $V = [P^{\rm op},\rm Set]$ its presheaf category with the Day convolution closed monoidal structure, which when $P$ is cartesian happens to also be cartesian. Now let $C$ be any locally presentable category on which $P$ acts through cocontinuous functors; the action then extends to an action of $V$ that has both adjoints, hence makes $C$ a $V$-category with powers and copowers, yet the enrichment is "unrelated" to any cartesian closure that $C$ started with and hence the latter will often not be $V$-cartesian-closure. In particular, $C$ could itself be a presheaf category $[J^{\rm op},\rm Set]$ with action induced from an action of $P$ on $J$.

This is great, but doesn't quite work for the case I'm really interested in when $V=[\Delta^{\rm op},\rm Set]$ is simplicial sets, since $\Delta$ is not cartesian monoidal. It could be generalized to that case since the cartesian closed structure of simplicial sets is, like any closed monoidal structure on a presheaf category, induced by a promonoidal structure on $\Delta$, so we could imagine a corresponding "pro-action" of $\Delta$ on some $J$ inducing a simplicial enrichment on $[J^{\rm op},\rm Set]$. However, I don't immediately see how to describe such a pro-action of $\Delta$ in concrete terms to make it easier to construct one.

So the question is, does there exist a complete and cocomplete category $C$ that is simplicially enriched with powers and copowers, and whose underlying ordinary category is cartesian closed, but such that the cartesian-closure adjunction is not simplicially enriched?

The answers to the other question certainly make it seem likely that the answer is yes. But note that there does exist at least one complete and cocomplete cartesian closed category $V$ such that every cartesian closed $V$-category is $V$-cartesian-closed — namely, $V=\rm Set$. Of course that is a very degenerate case, but it means that constructing a counterexample for any particular $V$, like simplicial sets, must use something about $V$ itself.

Answer 1

$\newcommand{\y}{\mathbf{y}} $Take $C = \mathcal{P}(a \stackrel{t}{\to} b) = \mathrm{Set}^{\cdot \leftarrow \cdot}$, so $C$ is freely generated under colimits by a morphism $\y t : \y a \to \y b$. Again I'll equip $C$ with the structure of a $V$-module, this time for $V = \mathrm{sSet}$. Suppose I know two (strong) monoidal left adjoints $A$, $B : V \to \mathrm{Set}$ and a monoidal transformation $\tau : A \to B$. Then define the action $- \odot - : V \otimes C \to C$ by $$K \odot \y a = A(K) \cdot \y a,$$ $$K \odot \y b = B(K) \cdot \y b,$$ with $K \odot \y t$ given by the composition of $\tau_K \cdot \y a : A(K) \cdot \y a \to B(K) \cdot \y a$ and $B(K) \cdot \y t : B(K) \cdot \y a \to B(K) \cdot \y b$, extending by preservation of colimits in the $C$ argument. Here $S \cdot X$ denotes the copower of $X$ by the set $S$. These formulas are functorial and colimit-preserving in $K$, so $\odot$ is an adjunction of two variables.

From the form of the action, we have an isomorphism between $K \odot (L \odot \y a) = A(K) \cdot (A(L) \cdot \y a) = (A(K) \times A(L)) \cdot \y a$ and $(K \times L) \cdot \y a = A(K \times L) \cdot \y a$ given by the strong monoidal structure of $A$, and similarly for $\y b$ and $B$; these isomorphisms are compatible with the map $\y a \to \y b$ because $\tau$ is a monoidal transformation. Extending by colimits, we obtain an associator isomorphism $K \odot (L \odot -) \cong (K \times L) \odot -$. Similarly the unit isomorphisms of $A$ and $B$ and their compatibility with $\tau$ provide a unitor $- \cong 1_V \odot -$, and (though I haven't checked the details) presumably the coherence conditions on $A$ and $B$ imply the coherence conditions making $\odot$ a monoidal action.

The map from $$K \odot \y a = A(K) \cdot \y a$$ to $$(K \odot 1) \times \y a = (K \odot \y b) \times \y a = (B(K) \cdot \y b) \times \y a = B(K) \cdot \y a$$ is given by $\tau_K \cdot \y a$, and so it is only a natural isomorphism when $\tau_K$ is.

Now luckily there are in fact two different strong monoidal left adjoints $\mathrm{sSet} \to \mathrm{Set}$ related by a monoidal natural transformation, namely $$A(K) = K_0, \quad B(K) = \pi_0 K,$$ with $\tau : A \to B$ the obvious natural transformation, which is not an isomorphism for $K = \Delta^1$.

One can compute the mapping spaces explicitly: for objects $X = (X_a \leftarrow X_b)$, $Y = (Y_a \leftarrow Y_b)$ of $C$, the simplicial set $\mathrm{Map}(X, Y)$ turns out to be the Čech nerve of the map $\mathrm{Hom}(X, Y) \to \mathrm{Hom}(X_b, Y_b)$. I'm not sure whether this description is very enlightening however.