$ \def\Set{\operatorname{Set}} \def\fP{\mathfrak{P}} \def\deg{\operatorname{deg}} $ Let $S \in \Set_{\Delta}$. Does there exist a model structure ( - which would be uniquely determined - ) on the category $(\Set_{\Delta})_{/S}$ having the following properties?:
- The cofibrations are the monomorphisms.
- An object $(X, p : X \to S)$ is fibrant if and only if $p$ is an inner fibration.
If it exists, it might be called the model category for inner fibrations over $S$ or the model category for correspondences over $S$.
The locally (co)Cartesian model category $(\Set_{\Delta}^+)_{/\fP}$ over the categorical pattern $\fP := (\deg_1(S),\deg_2(S),\emptyset)$ over $S$ comes pretty close and seems to me to be a marked version of what I'm looking for.