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Recall that a marked simplicial set is simplicial set $S$ together with a fixed subset of $S_1$ containing all degeneracies.
Recall that the category of “marked simplices” is the category of pairs $([n],W)$ where $W$ is a class of morphisms of $[n]$ containing all identities with no further restrictions. We require simply that morphisms between relative simplices preserve weak equivalences. Call this category $\Delta_m$.
We can define a functor $\Delta_m\to sSet^+$ in a somewhat obvious way by sending a marked simplex to the corresponding marked simplicial set. This yields a realization-nerve pair $Psh(\Delta_m)\leftrightarrows sSet^+$. Why is this adjunction not an equivalence of categories?
If we take the subcategory of $\Delta_m$ where we require the morphisms to be “strict on weak equivalences”, that is, the lluf subcategory where the morphisms are strict on markings, i.e. a morphism $f:([n],W)\to ([m],W')$ of marked simplices is called strict on weak equivalences if $f(W)=f([n])\cap W'$, is there still a chance of realizing $sSet^+$ as a presheaf category on this category?
If we further restrict $\Delta_m$ by requiring that morphisms are saturated, ($f^{-1}(W')=W$) are we still screwed?
Why is this adjunction not an equivalence of categories?
Because $sSet^+$ is not a presheaf topos, but a quasi-topos: the category of separated presheaves on a category $\Delta^+$ obtained from $\Delta$ by adding a single object $[1/2]$ and factoring $[1] \to [0]$ through this object.
Currently this is described at model structure for Cartesian fibrations in the section Cartesian closure. But I’ll move the discussion to a dedicated section now.
Dear Urs, are you familiar with a proof that $sSet^+$ is not a presheaf topos?
Dear Urs, are you familiar with a proof that $sSet^+$ is not a presheaf topos?
It is not a balanced category: the canonical morphisms $X^\flat \to X^{#}$ are mono and epi, but not iso.
mhm. Are you familiar with any attempts to find two quillen-biequivalent (to the cartesian and cocartesian model structures) model structures on a presheaf topos that’s ever-so-slightly bigger than $sSet^+$?
Are you familiar with any attempts
Haven’t heard and haven’t thought about it, no.
What do you mean by “biequivalent”? Do you want to consider $sSet_{Joyal}$-enriched model categories (“$\infty$-bicategories”)?
By the way, what did you mean in your very first message by
We require simply that morphisms between relative simplices preserve weak equivalences.
Didn’t you say in the sentence before that this is supposed to be a category whose objects are all simplices with markings? Where is the relative simplices and what notion of weak equivalence do you have in mind?
By weak equivalences, I mean the same thing as markings, sorry for the confusion.
By biequivalent, I mean that we have left and right quillen equivalences, that is, “bidirectionally quillen equivalent”.
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