Abstract classes of anodyne maps (relative to an interval) in a presheaf category are stable under smash products with monomorphisms?

Question

Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the small object argument such that $LLP(RLP(M))$ is exactly the class of all monomorphisms of $X$. Recall that a separated segment (a separated interval) is a triple $(I,\partial^0,\partial^1)$ where $\partial^i:*\to I$ (where $*$ denotes the terminal object) and such that the pullback of the diagram $\partial^0:*\hookrightarrow I\hookleftarrow *:\partial^1$ is the empty presheaf.

This triple defines a functorial cylinder $(I\times(-),\partial^0\times id_{(-)}, \partial^1\times id_{(-)},\sigma\times id_{(-)})$ where $\sigma:I\to *$ is the terminal map. By abuse of notation, we will write for any object $P$ in $X$, $\partial^0_P:=\partial^0\times id_P, \partial^1_P:=\partial^1\times id_P$ and $\sigma_P:= \sigma\times id_P$. Since $X$ is a presheaf topos, we can see easily that given any monomorphism $K\to L$ in $X$, the square:

$$\begin{matrix}K&\hookrightarrow &L\\ \downarrow&&\downarrow\\ I\times K&\hookrightarrow&I\times L\end{matrix}$$

(where the vertical arrows are the induced maps $\partial^0_K$ and $\partial^0_L$, or $\partial^1_K$ and $\partial^1_L$) is cartesian and is composed exclusively of monomorphisms. Because of this very fine property, we may define $I\times K \cup \{i\}\times L$ to be the subobject of $I\times L$ given by the apparent inclusion of the pushout where $i$ depends on the $\partial^i$ appearing in the above diagram. Adding to our list of suggestive notation, we define the map $\partial I:= *\coprod *$ considered as a subobject of $I$ by the canonical map from the coproduct $(\partial^0,\partial^1)$ (similarly, we define $\{0\}$ and $\{1\}$ to be the subobjects corresponding to the obvious maps (in this notation, $\partial I = \{0\}\coprod \{1\}$). We denote the previously mentioned inclusion $(\partial^0,\partial^1)$ by $b:\partial I\hookrightarrow I$, and as with the other distinguished maps, putting a subscript gives the obvious piece of the natural transformation.

Given any two morphisms $f:A\to A',g:B\to B'$ in $X$, define their smash product $f\wedge g:A\times B'\coprod_{A\times B} A'\times B \to A'\times B'$. Note that the smash product gives a monoidal product on $Arr(X)$ (the unit being the inclusion of the empty presheaf into the terminal one).

Given a separated segment $I$ in $X$, define a class of anodyne morphisms relative to $I$ to be a class $An$ of monomorphisms of $X$ satisfying the following three conditions:

$An_0:$ There exists a small set $S$ of monomorphisms such that $An=LLP(RLP(S))$.
$An_1:$ For any monomorphism $f:K\hookrightarrow L$, the smash products $\partial^i\wedge f$ are elements of $An$ for $i=0,1$.
$An_2:$ For any $f\in An$, the smash product $b\wedge f$ is an element of $An$ (recall again that $b:\partial I\to I$ is the canonical inclusion).

Question: Given any category of presheaves $X$, any separated segment $I$ on $X$, and any class of morphisms $An$ anodyne with respect to $I$, is it the case that given any monomorphism $f$ in $X$ and any anodyne morphism $g$ in $An$ that $f\wedge g\in An$? If this is true would you mind sketching a proof?

Answer 1

This condition is equivalent to the following condition:

The class $\mathrm{An}=\mathrm{An}(\Lambda_I(S))$ of anodynes generated by a small set $S$ of monomorphisms with respect to the cylinder $I$ is closed under finite Cartesian products with objects of $X=\mathrm{Psh}(A)$. That is, for any anodyne $f:K\hookrightarrow L$, and any object $B$ in $X$, the map $B\times f: B\times K\hookrightarrow B\times L$ is also anodyne, or equivalently, that the $A$-localizer $W(An)=W_I(S)=W(\Lambda_I(S))$ (which is accessible since $S$ is a small set) generated by $An$ is a Cartesian localizer.

This follows from Corollary 1.3.58, which in particular, says that $An$ has the following property:

If $u,v$ are composable monomorphisms in $X$ with composite $vu$ such that $u\in An$ and $vu\in An$, then $v\in An$.

Suppose that $f:K\hookrightarrow L$ is anodyne and $g:B\hookrightarrow C$ is any monomorphism.

We have the evident diagram:

$$\begin{matrix}B\times K&\hookrightarrow &C\times K\\ \downarrow&&\downarrow\\ B\times L&\hookrightarrow&C\times L\end{matrix}$$

In which the vertical maps are anodyne (by assumption).

Pushing out with respect to the top left corner, we get another diagram:

$$\begin{matrix}B\times K&\hookrightarrow &C\times K\\ \downarrow&&\downarrow\\ B\times L&\hookrightarrow&P\end{matrix}$$

and a unique map $P\hookrightarrow C\times L$ making things commute as they should. Notice now that $$C\times K \hookrightarrow C\times L = C\times K \hookrightarrow P \hookrightarrow C\times L.$$ However, the LHS is anodyne, and the map $B\times K\hookrightarrow P$ is anodyne as well, being a pushout of an anodyne map. Restating this, we see that $vu$ is anodyne and that $u$ is anodyne. Therefore, it follows that $v=f\wedge g:P\hookrightarrow B\times L$ is itself anodyne.

Also, it turns out that this answer holds in the slightly more general case whenever $I$ is a "good functorial cylinder" for $X$.

As the localizer $W(An)$ is accessible (by the axioms for a class of anodynes) and therefore is the class of weak equivalences for a model structure on $X$ where the cofibrations are the monomorphisms, this is equivalent to asking that the associated model structure is Cartesian-closed, since it implies that the fibrant objects are precisely those objects $B$ such that for every anodyne morphism $K\hookrightarrow L$, the induced map on internal function objects $B^L\to B^K$ is a trivial fibration in the associated model structure.