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I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.
For the record, I found the inductive characterisation in Cisinski’s book [Les préfaisceaux comme modèles des types d’homotopie, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [Notes on simplicial homotopy theory], but they take it as a definition.
I have added full publication data and section-pointers to a bunch of the references we had here, and cleaned up the list a little (slight re-organization, slight tweaking of wordings, for streamlining). For instance, it seemed to make no sense to list the proof by Goerss&Jardine not among the list including those by Gelfand&Manin and by Joyal&Tierney.
Then I added those further references we had at classical model structure on simplicial sets but not here yet.
In fact it’s a pain that I should go now to do the same updating there. So this means that instead I should make this references-section a stand-alone entry to be !include
-ed here and there. Will do this now at:
Add a reference later:
Title: A horn-like characterization of the fibrant objects in the minimal model structure on simplicial sets Authors: Matthew Feller Categories: math.CT math.AT Comments: 16 pages. Comments welcome! MSC-class: 18N40, 18N50 \ We show that the fibrant objects in the minimal model structure on the category of simplicial sets are characterized by a lifting condition with respect to maps which resemble the horn inclusions that define Kan complexes.
Interesting. Do you have an intuition for what these minimal fibrant simplicial sets are in the sense of generalizations of quasi-categories? I mean, is there a good higher category theoretic way to think of what $RLP(\text{iso-horns})$ is modelling?
Re #5: Good question! Maybe something close to (∞,∞)-categories? It looks like simplicial sets can model (∞,n)-categories for arbitrary n≥0. (n=0: Kan–Quillen, n=1: Joyal, n=2: Lurie’s new model for ∞-bicategories, etc.). So the minimal model structure should absorb all these. The lifting condition that he identified looks like a very natural one.
If Feller’s 2019 post at mathoverflow is accurate, the horn $\Lambda^2_1$ is Lawvere fibrant (I’m naming the fibrations after the exact cylinder, which is induced by the ’Lawvere segment’), so it doesn’t have any simplices we can interpret as a composite of its two arrows. So, I think it’s too optimistic to think Lawvere fibrant simplicial sets are higher category like.
Do simplicial sets really work as a model for higher categories when $n \gt 2$? I thought I had read it only works out fully when $n \leq 2$ and there are issues with $n\gt 2$, but it was a passing comment.
Re #7: I haven’t seen any arguments that would it is impossible for n>2 and would be very interested in seeing one. On the other hand, I no longer recall if/when/where I heard that simplicial sets can model (∞,n)-categories for n>2. Lurie refers to the work of Verity on complicial sets when he discusses simplicial sets as ∞-bicategories.
Ross Street’s original proposed definition of $(\infty,\infty)$-categories was based on simplicial sets.
I guess I’m taking the point of view that the $(\infty, n)\! Cat$, viewed as a tower of $(\infty, 1)$-categories, is correctly defined as starting with $\infty \! Gpd$ and iteratively taking the category of complete segal spaces (relative to the inclusion of groupoids), e.g. as described at category object in an (infinity,1)-category.
So, to make it precise, the question is whether or not these categories embed (and in a consistent way) as full subcategories of a relevant $(\infty, 1)$ category, such as the one presented by the model category mentioned in #4.
Is there an actual model structure on $sSet$ that presents $(\infty,2)\! Cat$? IIRC in (∞, 2)-Categories and the Goodwillie Calculus I, Lurie shows that there is such a model structure on scaled simplicial sets, but I recall looking for and never finding a claim that there is a corresponding model structure on $sSet$ that presents the same $(\infty,1)$ category.
(IIRC the Kerodon implies something about viewing a full subcategory of sSet as a quasicategory-enriched category of (∞, 2)-Categories; I think the hom-objects turn out to be the categories of lax functors, but I don’t recall what precisely can be said)
Can we maybe get an intuition for what the iso-horn conditions say in terms of composition of morphsims in a would-be higher category?
To start with, can we easily express lifting against inner horns as lifting against certain iso-horns?
(For a moment I though this would be evident, though maybe it’s not. But I haven’t really paused to think about it in any detail.)
My initial impressions are that isohorns inclusions are more like boundary fillers than horn fillers and wants to interpret the model structure as wanting to identify indiscrete spaces with the point.
At first look, my intuition wants to think of the “widening” construction as a coherent way to replace a vertex of a simplicial set with the indiscrete space on two points.
Suppose you take a sphere $\partial \Delta^n$, and you widen one of its points, say $k$, to get $W_k(\partial \Delta^n)$. This has two copies of $\partial \Delta^n$, corresponding to which of the two halves of the widened point it’s incident with.
If you fill in one of the two copies of $\partial \Delta^n \subseteq W_k(\partial \Delta^n)$, you get an isohorn. If you include a filler before widening, you get the isoplex $W_k(\Delta^n)$.
So, a simplicial set $X$ is Lawvere fibrant iff, for any widened sphere $W_k(\partial \Delta^n) \to X$, if one of the two copies of $\partial \Delta^n \subseteq W_k(\partial \Delta^n)$ has a filler, then the widened sphere itself has a widened filler.
On a related note, an interesting thing is that if a map $X \to Y$ is an inner anodyne map of simplicial sets, then $X_0 \to Y_0$ is an isomorphism.
I sort of view the description of the categorical equivalences as decomposing into two parts:
A lot Higher Topos Theory works with inner anodyne maps and inner fibrations, rather than the classes of Joyal trivial cofibrations and Joyal fibrations. It’s almost sort of like that’s the algebraic part of the model where all the interesting category theory lies.
I’m pretty sure this vague point of view on categorical equivalences is a large reason why I keep wanting to view the minimal model structure on $sSet$ as being primarily about working with simpicial sets in a way where vertices are only defined ’up to trivial choice’, and that category-theoretic aspect is an orthogonal concern.
There’s not going to be a model structure on $sSet$ that presents $(\infty,2)Cat$ if by the latter you mean the usual one where the functors are pseudo, because invertibility of a 2-morphism in the nerve of a 2-category is not preserved by simplicial maps. This is true already for (2,2)-categories; the Duskin nerve captures lax functors rather than pseudo ones.
Of course, asking whether there is such a model structure is a rather more specialized question than asking generally whether simplicial sets “really work” as a model for $\omega$-categories.
I have not looked yet at the recent paper on minimally fibrant simplicial sets, but my memory from when I thought about the minimal model structure a while ago was that minimally fibrant simplicial sets felt like $\infty$-groupoids (Kan complexes) equipped with additional data of not-necessarily-invertibly cells, such that the cells could be transported along (or “composed with”) the invertible morphisms in the $\infty$-groupoid, but not composed with each other.
I’d been somewhat curious about the minimal model structure on a general presheaf category, since they seem to be a natural thing to construct from a topos. It would be interesting if it turned out to actually model the category of $\infty$-presheaves, or at least a reflective subcategory thereof.
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