Homotopy Theory

2:00 AM

@AaronMazel-Gee I think the $\infty$-group completion of the monoidal groupoid generated by an idempotent object and an automorphism of that object is contractible. (I can't tell if this is related to @kiran's comment.)

2:12 AM

Note that the group completion map in this case is not injective on $\pi_0$.

2:25 AM

Oh, I forgot to add the relation $f \otimes 1_x = f = 1_x \otimes f$, where $x$ is the idempotent object and $f$ the generating automorphism of $x$.

No, scrap all of the above, I made a mistake in my calculations. D'oh.

Actually, if you change the relation to $f \otimes 1_x = 1_x = 1_x \otimes f$ maybe it's ok?

A: Why presheaves with transfer?

3:21 AM

(Though probably it's better only to add one of those relations.)

2 hours later…

NeverEnoughTime

5:21 AM

I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the analogue of $SH$ is the category $Sp$ of spectra. There are plenty of ways to define it, so let ...

Denis Nardin

6:16 AM

@AaronMazel-Gee Pick your favourite weakly contractible ∞-category, e.g. something with an initial object. I think if you have a cancellative monoid, as in your example, you might be able to say more though

6 hours later…

NeverEnoughTime

12:10 PM

I thought maybe an answer would say something about weil cohomology theories and pullbacks over correspondences $Z\subset X\times Y$, $[Z]=\alpha$, and $H^*(Y)\to H^*(X)$ given by $a\mapsto pr_{1,*}(pr_2^*(a)\cup \alpha)$ etc. I don't know much about this yet though

If you had any new take on that question now Denis, I'd be interested in hearing it

5 hours later…

5:04 PM

For any arbitrary monoid M, is the simplicial set BM a Kan complex? I feel like it shouldn't be, unless M is a group.

Where here BM is the usual simplicial object {...M×M×M⇶M×M⇉M}

I'm thinking that the above thing should be exactly the nerve of the category with one object and M as morphisms.

Therefore a quasicategory but not a Kan complex.

Yeah this must be true....

Denis Nardin

@JonathanBeardsley You are right: it's a Kan complex iff M is a group (it is always a q-category). Lifting $\Lambda^2_0$ gives a left inverse and $\Lambda^2_2$ gives the right inverse

Right, okay cool. Thanks. :)

Really vibing on the fact that when M is commutative you can get this thing by just thinking of M as a discrete Γ-space and restricting along the inclusion Δ^{op}→Γ^{op}

2 hours later…

Q: What is the coskeleton tower of a quasi-category?

6:43 PM

Back to an earlier question: does anyone have a good intuitive sense for what it looks like when you take an arbitrary simplicial set and 1-coskeletonize it? I.e. restrict to the full subcategory on the objects [0] and [1], and then right Kan extend back to sSet?

If I started with a quasi-category I guess I feel comfortable with guessing that I end up getting something like the ordinary category just on the objects and 1-morphisms, but having forgotten all higher information.

Oh this is sort of helpful:

I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely, this tower is. It appears to be homotopy-invariant, and each $k$-coskeleton looks like it is s...

Especially the part where Tim says "if you put something in which isn't a quasi-category then you have no idea what coskeletonization will do."

Ah actually.... hm. This seems to be saying that the 1-coskeletonization is actually a (0,1)-category? That's not good...

Maybe the point here is that all you're really remembering are which vertices are in the same connected components when you only remember [0] and [1].

7:11 PM

Here's an example showing why 1-coskeletonization is "badly behaved" on arbitrary simplicial sets. Let X be the simplicial set consisting of vertices 0, 1, 2, 3, and 1-simplices 0->1, 1->2, 2->3, 0->3. Then X is already 1-coskeletal because there's nowhere to attach a 2-simplex. Now, form Y from X by gluing in an inner 2-horn along 0->1 and 1->2. Then the 1-coskeletalization of Y consists of Y plus a 2-simplex with vertices 1,2,3.

So, while the map X -> Y is an inner anodyne extension, the map between their 1-coskeletalizations isn't even a Kan weak equivalence.

Whoops--the extra simplex in the 1-coskeletalization of Y should have vertices 0,2,3.

Am I drawing this correctly?

Hm where'd the photo go...

I don't know :) but my mental picture is a trapezoid with 0->1, 1->2, 2->3 along the top, 0->3 on the bottom, and the diagonal 0->2 drawn in

X is just the boundary, while Y is the boundary plus the top triangle, and the 1-coskeletalization of Y is the whole trapezoid

Ah no, I wanted to glue the simplex to X along the existing edges 0->1 and 1->2

You could think of the whole example as living inside Delta^3.

Oh, I see. I misunderstood what you meant by "glue in an inner 2-horn" I think.

So you're saying, I think, that you want to glue in another edge 0→2?

And the 1-coskeletonization forces you to "fill in" that triangle?

7:25 PM

I want to glue in 0->2 along with the 2-simplex with vertices 0,1,2, forming Y. Then the 1-coskeletonization forces me to fill in the other triangle.

Ah I see. You're gluing in an entire 2-simplex.

Yes, right, so X and Y are "equivalent" but then when you are forced to fill in that other triangle, you're "filling in the hole" that still exists in X.

Oh I see, right sorry. I was referring to the whole generating acyclic cofibration as a "horn" for some reason.

Right, so the fact that coskeletalization fills in the hole in Y but not the bigger hole in X makes it not homotopy invariant.

Great, thanks that was helpful to work through!

Hm, so can we think of cosk₁ in general as "filling in all available k-cycles for k>1?"

I guess that's what Prop 2.1 is saying here: ncatlab.org/nlab/show/simplicial+skeleton#general

In your example, nothing would have changed if you switched the edge 0→3 to being 3→0 right? I mean, there's still no place to glue in a 2-simplex, right?

X would still be 1-coskeletal

Yeah okay, I thought so.

Cool.

7:36 PM

but Y would also be 1-coskeletal, because there wouldn't be a place to attach a 2-simplex to Y

Right of course. Thanks.

You can think of X as encoding generators for a quasicategory, and the "hole" in X as representing two different 1-morphisms from 0 to 3 in this quasicategory. On a quasicategory, cosk_1 truncates the mapping spaces to subsingletons, i.e., it forces any two parallel morphisms to become equal. But it can't possibly do that for a general simplicial set because (for example) it only looks at a bounded amount of the simplicial set to decide where to add a 2-simplex.

To get the quasicategory presented by X, we should take a fibrant replacement of X, and we can think of X -> Y as the first step in such a fibrant replacement.

In this case this single step was already enough to "correct" the value of cosk_1 on X.

Oh, and by "looks at a bounded amount" you mean that it only fills in places where you've got parallel morphisms of exactly one edge and exactly two edges.

Right

It can't globally look and say "There's an edge here that wants to be a triple composition, so I should fill this in."

That's great. That's extremely helpful.

Ah great okay, and fibrant replacement in qCat is precisely this more global filling-in procedure.

1 hour later…