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Recall that a relative poset is a pair $(P,W)$ where $P$ is a poset and $W$ is a lluf subcategory containing all identities. A morphism of relative posets $(P,W)\to (P',W')$ is a relative functor, that is, a functor $P\to P'$ whose restriction on $W$ factors through the inclusion of $W'$. We denote the category of relative posets by $RelPos$.
Let $\mathcal{R}$ denote the full (and skeletal) subcategory of $RelPos$ consisting of relative simplices, that is, the full subcategory whose objects are finite nonempty linearly ordered relative posets. The category of presheaves on this category is called the category of relative simplicial sets.
We define a functor $T:\mathcal{R} \to sSet^+$ sending a relative simplex to the obvious marked simplicial set. That is, $([n],W_n)\mapsto (\Delta^n, Nerve(W_n)_1)$.
This lifts to a colimit preserving functor $|\cdot|_T:Psh(\mathcal{R})\to sSet^+$ (this is just the colimit over the marked realization of all relative simplices of a relative simplicial set) with an obvious adjoint functor defined by $Sing_{T}(X,\varepsilon)_{[n],W_n}=Hom_{sSet^+}(|([n],W_n)|_T, (X,\varepsilon))$.
I am pretty sure that $|\cdot|_T$ preserves monomorphisms (at least it should send them to cartesian cofibrations), and if this is the case, then according to a result of Cisinski, I can pull back Lurie’s class of cartesian equivalences to $Psh(\mathcal{R})$ if a certain accessibility condition holds, and this will give a left-proper combinatorial model structure on $Psh(\mathcal{R})$ with cofibrations the monomorphisms and weak equivalences the preimages of cartesian equivalences. If this is the case, I’d very much like to prove that the pair $(|\cdot|_T,Sing_T)$ is a Quillen equivalence.
However, before I undertake this task, I figured I’d ask you guys if you could think of any obvious counterexamples.
Why would I want to do this? First, a presheaf topos presentation of a model category where the cofibrations are exactly the monomorphisms is definitely extremely nice to have. Further, there are some very nice constructions on the category of relative posets that would generalize some of the nice features of the classical case. Lastly, such a model structure would also allow us to embed the classical case directly.
Hi Harry,
I haven’t thought about this, besides briefly last time when we talked about this and you had indicated already the inclination to find an equivalent model structure supported on a topos.
The idea with the relative posets certainly sounds very natural. After all, the markings in the marked simplicial sets are there to remember which morphisms ought to be equivalences, so are a simplicial version of the notion relative category as defined by Barwick and Kan and discussed at model structure on categories with weak equivalences. Your category of relative posets is precisely the full subcat of their category of relative categories, restricted to posets. That’s probably not a coincidence.
So I would find it quite interesting if you could establish the Quillen equivalence that you indicate you are after. Unfortunately, apart from that I don’t currently have any technical details to offer and am unlikely to have at least before the weakend.
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