Homotopy Theory

Jonathan Beardsley

00:38

Dunno if people know about it, but there is a "category theory" chat room now. I guess most of us don't really have any reason to go there since we basically talk about category theory in here without any real issue... but in case anyone's interested: chat.stackexchange.com/rooms/68360/category-theory

Harry Gindi

01:01

I just saw a talk by Thomas Nikolaus yesterday where he talked about the (infinity,1)-categorical Day convolution. I feel like the boundary between the two subjects has pretty much vanished

By the way, has anyone else looked through all those papers by Verity-Riehl?

Aaron Mazel-Gee

@HarryGindi depending on the context, i don't think it's generally appropriate to try to prove everything in a talk anyways (though my time in germany suggested that people there think otherwise)

Harry Gindi

the framework is so much nicer than the setup in HTT

@AaronMazel-Gee I had notes with full proofs, I just completely botched the actual presentation and 'blackboard performance' part of everything

I need practice, I think that's really the problem

Aaron Mazel-Gee

well sure, all of us could always use more practice. but my point is about the point of talks in the first place

Harry Gindi

Yeah, but without the proofs, my assigned talk had no content

I had to present HA 6.1.6

there is only one proposition in the whole section

Aaron Mazel-Gee

ah, okay

even for those types of talks, though, i still often find it enlightening and helpful to restructure the talk in comparison with the original presentation

giving a top-down approach, instead of a linear one

Harry Gindi

01:06

I gave that a shot, but then Justin Noel told me to define the norm map before continuing

Aaron Mazel-Gee

in communicating verbally you're "allowed" to be so much more expressive about intuition than in writing

haha well if that's the main topic, then what were you planning to say about it without the definition?

oh, but it takes a few minutes to define, doesn't it -- the iterative thing where you keep asking for the previous norms to be equivalences so you can define the next one

Harry Gindi

yep

Aaron Mazel-Gee

yeah, in that case i'd want (i) an illustration with abelian groups / chain complexes, (ii) the big-picture intuition, and only finally then (iii) the official definition

Harry Gindi

and you also have to prove at each stage, at least in the presentation in HA, that you can check it on its fibres

it's kind of silly though because in the main proof it's literally only used in the case where Y is Δ^0 and X is Σ_n

or rather, BΣ_n

Aaron Mazel-Gee

i mean, that construction is a cool thing in and of itself, but going through it without knowing "why" is probably kind of painful

Harry Gindi

01:11

Yeah, I didn't understand it until Uli Bunke asked a question and Thomas Nikolaus said it's useful because the tate construction often vanishes

Aaron Mazel-Gee

wait, is thomas there? or just visiting?

Harry Gindi

not sure, he gave a talk at the higher invariants oberseminar, and he was in the audience at my talk for the higher categories seminar

I think maybe he's visiting? I am not sure though

I haven't met everyone in the department yet

Aaron Mazel-Gee

regensburg probably has the largest homotopy theory group in the world, doesn't it?

Harry Gindi

I think it's maybe 70% homotopy theorists?

If you include Cisinski's motivic group as homotopy theorists instead of geometers

Aaron Mazel-Gee

that's great, but also it's 70% of a large number

if they use homotopical methods, i'd say they belong under the homotopy theory umbrella at least

Harry Gindi

01:16

I mean, Cisinski started in homotopy theory at least

he's a homotopy theorist for sure, I just haven't been to the motivic sheaves obserseminar

I wanted to go, then I found out that they're already a year and a half in

so is the higher categories seminar, but it's slightly more elementary; we're doing the Goodwillie Calculus in infty-categories this semester

the Oberseminar I think is current active research

Now that I at least gave my talk for the HC seminar, I can start writing some stuff I've been thinking about and present it at the AG-seminar, hopefully with a better presentation this time

Dom Verity told me that his stuff with Emily Riehl all generalizes straightforwardly to complicial sets, so I started working it out and I think I understand at least the components needed to also get theta-sets and theta-spaces working with it

the main confusing part seems to be figuring out what the 'lax homotopy-coherent nerve' in the complicial case looks like for theta-sets

also, Dimitri Ara and I are still unsure if the lax tensor product does extend to a genuine biclosed monoidal structure on theta-sets, or if we will have to use the same trick that Moerdijk and his collaborators used to get the BV tensor product of dendroidal sets to work

(build a biclosed (but not associative) bifunctor and show that it is homotopy-equivalent to a multitensor when mapping into a fibrant object )

anyway thanks for the advice

going to bed

Bogdan

09:05

@AaronMazel-Gee Thomas is still at MPI, but he's going to be in Munster starting from March or so

Hopefully his replacement here will be again somebody in homotopy theory :)

Bruno Stonek

10:05

where can I find a study of the model category of $\mathcal{L}$-spaces ($\mathcal{L}$=linear isometries operad)? it's a hard thing to google

what I really want to know is if a CW complex which is a topological commutative monoid with cellular multiplication gives rise to a cofibrant $\mathcal{L}$-space

Bruno Stonek

10:56

hmm, I guess that shouldn't be true? $\mathcal{L}$ is a cofibrant replacement for $Com$, the map $\mathcal{L}\to Com$ induces a left Quillen functor from $\mathcal{L}$-spaces to topological commutative monoids. that's... not what I wanted

Bruno Stonek

11:15

but maybe the restriction functor $Alg(\mathcal{L})\to Alg(Com)$ is not only right Quillen but also left Quillen? in other words, is there a "coextension of structures" functor?

so the general question would be: if $\mathcal{O}\to \mathcal{O}'$ is a morphism of operads, is the "restriction of structures" functor from $\mathcal{O}'$-algebras to $\mathcal{O}$-algebras both right -and left- Quillen? (ouch, I inadvertently started a monolog)

Rune Haugseng

12:15

No, it will typically not preserve coproducts (think about the coproducts of associative versus commutative algebras). It does preserve sifted (homotopy) colimits though.

Justin Noel

@Harry: You really shouldn't be so hard on yourself. I would be happy to have you talk again in the seminar.

Everybody has to give a first talk and a prerequisite for giving a very good talk is being comfortable giving talks. I think for most people, this just takes time and practice. As long as you try to get better and you think about what works and what doesn't, you will get better.

From my experience, mathematicians are much happier with hand-waving in proofs (in talks) than they are with hand-waving through definitions. We are partially capable of filling in details in proofs, but wi…

@BrunoStonek: You can find some material on L-spaces in \S 5.2-5.3 here (arxiv.org/pdf/0810.4535.pdf).

Harry Gindi

@JustinNoel I didn't practice and it showed

I sort of expected it to go better is all haha

I was working last night on some stuff that I'd like to present at Uli Bunke's seminar, and there is some kind of deep combinatorics that I don't think anyone else has really looked over

I don't understand it well at all

I have other stuff I could present where I've actually proven things, but I think presenting the framework of this thing and the motivation behind this problem is going to be really important moving forward

so I don't know which seminar would be appropriate

Bruno Stonek

12:34

@RuneHaugseng oh, of course, thanks! as I said, I'd like it to preserve cofibrant objects, but I guess this is not true in general. I have no intuition on what is a -cofibrant- $\mathcal{L}$-space

@JustinNoel thanks! I'm having a look

Harry Gindi

I don't know if it's appropriate to set out the results of some other mathematicians at an AG seminar

Justin Noel

Regarding Regensburg and homotopy theory: Amongst the professors we have Bunke, Cisinski, and Naumann. Amongst the assistants there's Alexander Engel, Adeel Khan, Markus Land, Georgios Raptis, and myself. We also have a number of postdocs including Daniel Schaeppi and Matan Presma.

The division between homotopy theorists and non-homotopy theorists is somewhat arbitrary here. There are also people like Georg Tamme, Florian Strunk, and Moritz Kerz who are using homotopy theory and prove results in this realm.

Harry Gindi

also Nat Stapleton is doing chromatic stuff

I know him from waaay back

Justin Noel

Of course there is Nat. However, he is leaving for Kentucky as of next week.

Harry Gindi

Ah, I didn't know he was leaving!

Justin Noel

12:42

Yes. It is sad.

Harry Gindi

Going to have to hit him up to get drinks before he leaves

Justin Noel

Although, I'm very happy for him.

Harry Gindi

Yeah, I don't know the details, but I know him from going to the midwest topology seminar and also through Eric

Justin Noel

Copenhagen is almost certainly bigger than us. Galatius, Grodal, Hesselholt, and Wahl and an awesome team of postdocs including Barthels, Clausen, Haugseng, Hausmann, and Sprehn.

Sorry, Tobias Barthel.

Harry Gindi

I'm kinda looking for someone to advise me, but I'm not sure who at the department is interested enough in infinity,n categories. I have maybe four or five open problems relating to Theta/cellular sets and fleshing out different aspects of the theory.

some of them I think are pretty accessible to me, others are quite difficult

if you have any recommendations who I should speak with, let me know

Justin Noel

12:51

@HarryGindi: Perhaps we should discuss such things in a more appropriate venue, such as in person.

Harry Gindi

Sorry, I'm just reluctant to waste your time in person

John Berman

17:14

I have a closed symmetric monoidal category C which I would like to localize at the unit. That is, I want to localize at all X\to Y such that C(Y,1)\to C(X,1) is an isomorphism. I would like to show that the localization retains a nice symmetric monoidal structure; it would be great if it were also a full subcategory of C. Are there nice conditions I can check that would guarantee this?

Denis Nardin

18:35

@JohnBerman Since the morphisms you're inverting are closed under colimits, I think you are looking for a Bousfield localization. In that case, you can check that the localization is compatible with the symmetric monoidal structure by seeing if the functors X⊗- send L-equivalences to L-equivalences

John Berman

I had hoped it would be that simple, but wasn't sure. When I say that C(Y,1)\to C(X,1) is an isomorphism, I mean as objects of C. Will that be an issue?

Also, since we are dealing with a closed symmetric monoidal structure, it is clear that X⊗- preserves L-equivalences.

Denis Nardin

I think everything works. I don't know a reference for 1-categories though if you need it (although technically the ∞-categorical statement implies the 1-categorical statement). Essentially you define x⊗_{LC} as L(x⊗_C y) and 1_{LC}=L1_C

John Berman

19:00

No, C is definitely an infinity category, and I would appreciate a reference for this context.

Drew

19:12

@JohnBerman 2.2.1.9 of Higher Algebra, I think

Harry Gindi

@JohnBerman Thomas Nikolaus gave a really nice talk about this at at the TC Seminar

he proved a generalized version of the infinity,1 version of the Day Convolution theorem

hey thermo @EricPeterson

Dylan Wilson

19:31

sorry but why does that localization even exist? C(-,1) is a colimit preserving functor to C^{op}. Usually the way I know how to show some localization exists is by saying I have an accessible functor to a presentable infty-category and the morphisms I'm inverting are the ones which become equivalences there (since equivalences in presentable (infty)-categories are generated by a set). But if C is presentable, C^{op} is usually not presentable.

Denis Nardin

That's a good point. We should probably check that the category of arrows we are inverting is accessible.

(since in that case we can check if an object is local on a generating set of the arrow category)

Shay Ben Moshe

20:21

Hey, I have another small question, this time about Morava E-Theory.
As far as I understand, there are many things that go under that name, or at least related (e.g. with coefficients Z_(p)[v_1,...,v_n,v_n^-1] as in Ravenel, or W(\bar{F}_p)[[u_1,...,u_n-1]][u^+-1] as for E_n^nr in Rognes, and some more).
My question is, are these spectra Bousfield equivalent? What is the relationship between them?
Thanks
edit: oh, and if anyone has a reference regarding the relationship between different ones, that would be great.

John Berman

20:38

@DylanWilson How is this issue resolved when we try to localize (spectra for example) at a cohomology theory?

skd

@ShayBenMoshe mathoverflow.net/questions/36345/…

Shay Ben Moshe

@skd thanks. I already saw that question earlier this week, but I don't think that it directly answers my question (unless I missed something), and unfortunately I don't understand enough details yet, to be able to understand if it implies what I'm asking somehow.

Dylan Wilson

20:59

@JohnBerman because usually we use the homology theory, right? Then you're inverting stuff which becomes an equivalence after smashing with E. That preserves colimits (so it's certainly accessible) and lands in Sp, which is presentable, so that implies the E-equivalences form an accessible subcategory of arrows.

John Berman

Sure but what is this? ncatlab.org/nlab/show/cohomology+localization

skd

it describes the relationship, at least. i think the bousfield classes of morava e theory and johnson-wilson theory are the same

and both have the same bousfield class as K(0) \/ ... \/ K(n)

this is because E-theory splits multiplicatively as a wedge of suspensions of L_K(n) E(n)

Shay Ben Moshe

I am familiar with the statement about K(0) \/ ... \/ K(n)

but I can't see where this (or something similar) is stated in that answer

skd

i don't think the statements about the bousfield classes appear in that answer

Shay Ben Moshe

and are you familiar with any reference to that?

Dylan Wilson

21:08

@JohnBerman well it says right there "if it exists", and then it goes on to reference some paper that says its existence is implied by some large cardinal axioms in set theory... but that's pretty close to cheating. The weak Vopenka's principle is I think equivalent to saying like "localizations of presentable categories always exist" or, rather, every subcategory closed under limits is reflective.

John Berman

So you suspect that the localization is pretty unlikely to exist?

Dylan Wilson

or rather, I think the existence of cohomological localization is in general independent of set theory...

John Berman

Hah that sounds reasonable to me!

Eric Peterson

21:41

@ShayBenMoshe yes, those spectra are bousfield equivalent as homology functors on spectra. there are a variety of senses in which morava E-theory is best behaved when considered internally to the K(n)-local category, where this is not an interesting question: the K(n)-local category has no localizing subcategories, so either you kill everything or you don't, and no variant of E_n (including L_K(n) E(n)) is that destructive

the differences between these spectra are largely 'unimportant', which is why they're under-discussed. given your favorite two variants of E-theory, there is typically some third variant of E-theory and a pair of collapsing Lyndon-Hochschild-Serre spectral sequences comparing the Adams spectral sequences for versions 1 and 2 with version 3, where you can see that the information in 1 and 2 are 'essentially the same', literally up to some frills that get deleted by the LHSSS

i'm not sure how to make this super transparent or where to point you. i was also confused about these kinds of things when i asked that MO question, and i don't know that i ever got truly un-confused or if i just got sufficiently used to the situation that i convinced myself that i could untangle this if i absolutely had to

i still think this 'work inside the K(n)-local category' maxim is important (cf. also chromotopy.org/topologized-objects ), at least as a starting point for thinking about these objects before you get dumped into the modern landscape of transchromatic phenomena

John Berman

22:07

@DylanWilson @DenisNardin Do we have any idea what happens if we localize spectra at stable cohomotopy or even localize abelian groups at Hom(-,Z)? I am realizing these are fairly aggressive localizations. In the abelian group case, we are killing off not only torsion groups but also Q.

On the other hand, this localization at Hom(-,1) doesn't do much to dualizable objects. In the case I am interested in, many of the objects are "almost dualizable" in the sense that they should become dualizable after localizing... if I can make sense of what the localization means.

Shay Ben Moshe

@EricPeterson many thanks for the detailed answer!
I will definitely try and think about what you said regarding working in K(n)-local spectra, in relation to E-theory, and I'll read that link. what you said about not having localizing subcategories, is just by the thick subcategory theorem, right?
if by any chance you happen to remember references for what you said about LHSSS, or anything in that spirit, I'd be happy to hear about that :)
thanks again

John Berman

In fact, maybe I can say something about my particular case, though I am not sure how helpful it will be. It is closely related to the following fact: localizing dgCat at Morita equivalences produces (k-linear) stable infinity-categories. This is a cohomology localization, in the sense that it is like localization at dgCat(-,Mod_k).

skd

23:43

@ShayBenMoshe for the statement about localizing subcategories: theorem 7.5 of hovey-strickland

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