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Stream: theory: algebraic topology

Topic: exact cylinders and Cisinski model structures

James Deikun (Dec 13 2023 at 14:10):

As a byproduct of my participation in the reading group on Cisinski's book I'm trying to develop a better understanding of the characterization of Cisinski model structures on presheaf categories and maybe a slicker presentation of the result.

James Deikun (Dec 13 2023 at 14:17):

Let the abstract interval category be the following model category:

• The underlying category is Psh($\mathcal{I}$), where $\mathcal{I}$ is the category with two objects 1 and I and three morphisms $\partial_{0,1} : 1 \to I$ and $\sigma : I \to 1$. (Because the Yoneda embedding is continuous, 1 indeed remains 1 in the presheaf category.)
• The cofibrations are generated by $\partial_{0,1}$ and their copairing, referred to as $\partial$.
• The acyclic cofibrations are generated by $\partial_{0,1}$ alone.

James Deikun (Dec 13 2023 at 14:41):

Is this actually a model structure? Well, you can view an object of the abstract interval category as a set of points and a set of intervals, so that each interval has two endpoints and each point has a "trivial" interval with itself as both endpoints. A map of presheaves is just a map from points to points and one from intervals to intervals that are consistent with these. The [[small object argument]] constructs a factorization system from each of these generators. What then remains to be seen is that these cooperate to produce a well-behaved notion of weak equivalence.

James Deikun (Dec 13 2023 at 16:00):

We take as weak equivalences those morphisms that have an inverse up to the equivalence relation generated by $I \times$-left homotopy. Unfolding this, it means that for any two points in the codomain whose images are connected by an interval, any preimage pair are connected by a finite zigzag of intervals in the domain; and furthermore, a finite zigzag in the codomain connects any point to one in the image of the map. In short: it is an isomorphism on weakly connected components. In fact, the connected components functor gives exactly the homotopy category for these weak equivalences; thus they obey 2-out-of-3.

From the lifting properties on the generating morphisms, one can see that acyclic fibrations are fibrations and weak equivalences. Also, since any pushout of $\partial$ lacks a homotopy inverse, a cofibration that is a weak equivalence is acyclic. Now let's take a fibration that is a weak equivalence, and factor it into a cofibration followed by an acyclic fibration. The cofibration must also be a weak equivalence by 2-out-of-3. This means that it is acyclic, but since it is factored out of a fibration it must be an iso and the original fibration must have been acyclic. And if we have an acyclic fibration, it must be a weak equivalence since any pushout of $\partial_i$ is a weak equivalence.

James Deikun (Dec 13 2023 at 16:01):

(There are some technical steps I left out of these proofs; do they seem basically solid?)

James Deikun (Dec 13 2023 at 16:13):

Anyway, define a weakly exact cylinder to be a bifunctor $\otimes$ from the abstract interval category $\hat{\mathcal{I}}$ and Psh(S) to Psh(S) which is the left adjoint of a [[two-variable adjunction]] and such that the [[pushout product]] of a cofibration and a monomorphism is a monomorphism. I'll try to develop this more in detail later, but it seems like this can probably replace the idea of an exact cylinder in the development and it will be possible to get the minimal model structure for a cylinder as basically a generalization of a [[transferred model structure]] from the above model structure along the cylinder and general Cisinski model structures via left localization of these.

Reid Barton (Dec 13 2023 at 18:22):

This is not going to be a model category. I am not sure exactly where the reasoning you gave goes wrong, but here is one way to see that there is a problem. The intersection of the cofibrations and the weak equivalences has to be the class of acyclic cofibrations. However, by the small object argument, all the acyclic cofibrations in your category have to be retracts of transfinite compositions of pushouts of the generators, and by ordinary simplicial homotopy theory (viewing your category as the full subcategory of $\le 1$-dimensional simplicial sets), all of these maps will actually induce isomorphisms not just on $\pi_0$, but also on the whole homotopy type (i.e., in this case also $\pi_1$).

Reid Barton (Dec 13 2023 at 18:25):

In my opinion a good perspective on this is:

• The object you wrote down (the category and its generating cofibrations and acyclic cofibrations) is the right object to consider, and you just have to live with the fact that it's not a model category (it's only a "premodel category").
• However, another useful thing to consider is the free monoidal premodel category on this object subject to the condition that $1$ be the unit object for the monoidal structure. That turns out to be the category of "plain" cubical sets. In this case we are lucky, and we actually do get a model category.

Reid Barton (Dec 13 2023 at 18:27):

I wrote some notes on a related topic a few months ago, which are not really finished, but here they are anyways:
notes.pdf

Reid Barton (Dec 13 2023 at 18:29):

Your "weakly exact cylinder" is very close to the notion of a "cylindrical premodel structure" which appears there; though you probably also wanted to require that $1 \otimes -$ be the identity functor.

Mike Shulman (Dec 13 2023 at 18:29):

Also relevant to the "universality" of the model category of cubical sets is https://arxiv.org/abs/1602.05313.

James Deikun (Dec 13 2023 at 19:50):

Thanks, these notes are helpful, and so is https://arxiv.org/abs/2004.12937 !

I see what the problem is with my argument for the "model structure" on the abstract cylinder now: the pushout of $\partial$ and the codiagonal has a homotopy inverse. This makes both arguments that weak equivalences are acyclic fall apart. I think it ultimately comes down to the fact that there's no way to distinguish the "filled" square from the "hollow" square and so the weak equivalences have to squash down too much for there to be an interesting model structure?

James Deikun (Dec 13 2023 at 19:56):

Also: I do want to require that $1 \otimes -$ is the identity functor, but I thought I got that for free for the same kind of reason that the global sections functor is unique. If not though I definitely should require it.

James Deikun (Dec 13 2023 at 20:07):

It seems like if I pass to the plain cube category the weak equivalences will just be wrong; the homotopy category of the "abstract interval category" should just be $\mathrm{Set}$ so that the enrichment-like part of the weakly exact cylinder composed with the localization will give the elementary homotopy classes of maps.

James Deikun (Dec 13 2023 at 20:26):

(Maybe this is an unnecessary level of concern and there's a way to justify just taking $\pi_0$ regardless of it not being the localization, but I don't have such an argument to hand.)

James Deikun (Dec 14 2023 at 13:27):

I'm beginning to think that in the cubical case, where $Y$ is fibrant $[X,Y]_{\otimes}$ is already homotopically a set, but I'm having a hard time proving symmetry in particular.

Reid Barton (Dec 14 2023 at 17:35):

James Deikun said:

It seems like if I pass to the plain cube category the weak equivalences will just be wrong; the homotopy category of the "abstract interval category" should just be $\mathrm{Set}$ so that the enrichment-like part of the weakly exact cylinder composed with the localization will give the elementary homotopy classes of maps.

They actually will be the correct weak equivalences, though you're right that a priori the characterization of the weak equivalences looks different.