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Recall the zigzag category $T$. It is known that $T$ is a reedy category. Suppose we weaken the definition a bit, by only requiring that “-” objects map to “-” objects, but not necessarily maintain the partition, and call these “weak zigzags”. Is there any existing work considering presheaves on this category? At least superficially, it seems like presheaves on the zigzag category resemble something like stratified simplicial sets, so I’m wondering if anybody has tried to equip them with a model structure of sorts.
As it involves stratified simplicial sets, have you checked through the papers and monograph of Dom Verity?
Hmm, my original assessment was wrong. More correctly, I should have said, “these resemble marked simplicial sets”. However, I just realized that perhaps we shouldn’t weaken the axioms for Zigzags, to avoid double-counting simplices.
Perhaps a better way to state this would be: Let Z be the category of pairs $([n],W)$ where $W$ is a lluf subcategory of $[n]$, and where the morphisms $([n],W) \to ([m],W')$ are functors $[n]\to [m]$ such that $im(W)=im([n])\cap W'$. Then take the category of presheaves over Z. That is, they’re not exactly zig-zags, but they’re not exactly relative posets either.
The idea here is, we define a functor $Z\to sSet^+$ sending $([n],W)\mapsto (\Delta^n,\mathfrak{N}(W)_1)$. Notice that these range over all of the ways we can mark $\Delta^n$ for each $n$, and by abstract nonsense, this gives us a colimit-preserving functor $Psh(Z)\to sSet^+$, whose right adjoint is a nerve functor sending a marked simplicial set to the $Z$-set of its edge-marked simplices. I’m hoping that it is an equivalence of categories, but I haven’t checked.
At first glance, it seems like it honestly should be.
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