@DenisNardin i'm glad this is being formally published! good talk, good notes

these handbooks are good projects and the few contributions to the handbook of homotopy theory I've read are excellent

I'm more familiar with the handbook of geometric topology, which is itself a fantastic reference that spawned a few canonical pieces of writing

@HarryGindi You were asking a while back about the proof that a morphism of n-quasi-categories is an equivalence iff it is eso and fully faithful. I've just scanned my notes from the seminar talk I gave on this last year: web.science.mq.edu.au/~alexc/20180926.pdf

Thanks, will check it out!

Does anyone know an easy way to prove an ∞-category is stable if I have no way of accessing the suspension functor?

What do you have access to?

So, the category is a Bousfield localization of an unstable category I control quite well. So I can easily compute loopspaces, and I know the suspension is given by localizing the suspension in the ambient category. Unfortunately the localization functor is what I don't want to compute

I also have a decent description of the generating equivalences, but I doubt it'll help

I can’t think of any nice tricks. It seems like you’d have to somehow show the unit or counit of loops-suspension in the ambient category can be built out of your generating equivalences, any way you slice it. I imagine it might be slightly easier to prove that the category is semiadditive first, but I dunno how helpful that is.

Oh everything is semiadditive. In fact the ambient category is prestable

I could actually be satisfied with just the fact that the localization itself is prestable (I only need that the map to the stabilization is fully faithful).

Are you able to prove that f is one of your new equivalences iff \Sigma f (in the ambient category) is? I think that might be enough.

Hrmm... I don't think I have a good criterion to decide when something is a local equivalence.

Hello

Given Lie groupoids $\mathcal{G},\mathcal{H},\mathcal{K}$ and morphisms $\phi:\mathcal{G}\rightarrow \mathcal{K}, \psi:\mathcal{H}\rightarrow \mathcal{K}$ we have what is called $2$-fibre product usually denoted by $\mathcal{G}\times_{\mathcal{K}}\mathcal{H}$... Under some conditions this is a Lie groupoid... In some places, this $2$-fibre product is called homotopy pullback... Can some one tell In what context homotopy is coming here?

a homotopy equivalence of categories is also called an equivalence of categories

@HarryGindi That's not the terminology I'd use :P

@PraphullaKoushik There is this general notion of homotopy limit and homotopy colimit that's been originally defined in homotopy theory but it applies in a much general setting

A special case of this setting are (2,1)-categories, i.e. 2-categories where all 2-morphisms are isomorphisms

yeah, it means that it has a homotopy-universal property in this case

The 2-pullback is a particular case of homotopy limit

An explicit model of a hopullback in Cat is also called an iso-comma

The connection with homotopy theory is that if you squint you can see that invertible 2-cells look sort of like homotopies of maps

I don't know how to 'define' a holim in model categories... suspicious =]

@HarryGindi I wasn't really thinking of model categories when writing the stuff above, but don't you?

Oh, I know how to 'define' it

as a derived functor

I was making the distinction between a definition and construction

There's a book of Dwyer-Hirschhorn-Kan-Smith that came out right before the ∞-category 'revolution', as it were

and they did come up with an intrinsic characterization without passing to localization

I just don't remember it because it was complicated

@HarryGindi What is homotopy-universal property here?

@DenisNardin By the way, when you guys did all the stuff with Clark about parametrized infty-categories

did you guys do a version of parametrized straightening?

I have a use for that in a different situation, and I was wondering what is the reference

@PraphullaKoushik This was exactly the the thing I didn't want to discuss. The whole concept of a homotopy-universal property is hard to formulate without either machinery or working in ∞-categories

@DenisNardin "The connection with homotopy theory is that if you squint you can see that invertible 2-cells look sort of like homotopies of maps" It looks like I understand but not sure... just like functors are like maps and natural transformation are like homotopies between these two maps... Is that it or is there some thing more happening here :)

@PraphullaKoushik there is a model structure on Cat where the homotopies are modeled as natural isomorphisms

@PraphullaKoushik That's all that's happening. But it's deeper than it looks: it turns out that both things are special case of a general set up where you can do things

@HarryGindi Yes. Proposition 8.3 in exposé I

Which paper?

I only saw the talk

arxiv.org/abs/1608.03657

thanks, buddy!

@HarryGindi Can you think of some basic lecture notes or some paper where this notion of Model structure is discussed.... I do not need it now, simply asking...

@DenisNardin Can you think of some paper/notes where this is explained in detail... As I said to Harry Gindi, I do not need this now, just curious...

It's a big big story

:D :D

From where did you learn homotopy theory first time?

One of the standard references for model categories is Hovey's Model categories. It's relatively quick.

Well, I did a bit of basic homotopy theory during my undergrad, but where I really learned homotopy theory was during a thing that's called the "Kan seminar"

Basically read at least two papers a week from a list of historically important papers. It's exhausting, but you learn a lot

I really like Hirschhorn better than Hovey as a reference

It has less basic examples

But I agree that it does the advanced theory better

but I never learned directly from a textbook, so I can't recommend it for that purpose

Yes, I heard about Kan Seminar... It is advantageous (may be stressful also) to be in such big places... You do not see anything close to that happening in India... :)

@PraphullaKoushik I don't know what stage of your career you're in, but if you're an undergrad, you might want to apply here at Regensburg for a master if you're really interested in homotopy theory

It's not very expensive (it doesn't even register compared to the US for international students)

@HarryGindi I am in my 3rd year of PhD in India.. :) :) The level is different (not great of course :D) here...

ah

Yeah, I don't know much about Ph.D. in India

I am trying to work hard to complete this and join as Post Doc in a better place..