Hello Espen Nielsen, I happen to think these days about topological categories and the behavior under various notions of classifying spaces, too. Initiated by your question about the colim-hocolim behavior, some questions, related to what you said, popped up in my mind. Are you interested in discussing these?

Hi there. Sure! I managed to royally embarrass myself with that question, but I I would very much like to discuss this with you.

I'm happy to hear the second and don't think you embarrassed yourself.

I wondered about the way you realize simplicial spaces. Which motivation do you have to go to bisimplicial spaces firstly?

This was based on the assumption that the geometric realization of the singular simplicial set is homotopy equivalent to the space itself, which should be true at least in the presence of mild regularity conditions.

Then using the fact that the diagonal realization of a bisimplicial set is homotopy equivalent to its usual realization.

I have to run to the store, be back in about an hour.

I see. Do you have any advantages of this way of realizing in mind instead of using the "regular one", i.e. take the (topological) nerve to get a simplicial space and use the definition of the regular geometric realization including the topologies of the levelwise nerves?

See you later

The only advantage is that I don't really like coends that much. I suppose there is not much of a technical advantage. I guess it's just about what you find more intuitive to work with.

Seems like I think more concrete in terms of the set-theoretical definition. The nlab page on geometric realization of simplicial spaces compares the different notions too, but you surely are aware of that.

Trying to define the colimit of topological categories intuitively (taking colimits of objects and morphisms in the category of spaces one works) the only problem seems to be to define the composition since fiber products do not commute with colimits in general, right? This seems to happen even for colimits over the natural numbers. (This is the case I'm mostly interested in, too.)

Right. However, it seems to work out in some well behaved cases. In particular, if all the functors are closed inclusions on objects and morphisms, and maybe with some more regularity on the categories, like the source and targets being fibrations, I think you can get it to work.

At least that's the impression I got from Peter May.

I think so, too. But you did not check it in detail neither?

I have only checked it for the categories I am interested in, in which I have an explicit description of all the maps involved and it was not hard to work out.

In a more general case, it seems an argument is necessary.

Basically it's seems to be just a question of colimits of topological spaces. I have the impression that if the natural map colim(morC_i (x_obC_i) morC_i)->colim(morC_i) (x_colim(obC_i)) colim(morC_i)) is a homeomorphism, the intuitive definition of the colimit category should really give the colimit in the category of topological categories.

Yes, definitely.

So in your case, you just checked that this map is a homeomorphism by hand?

Yes.

It may be the case that accidental colimits exist, in which the colimit category is something unintuitive though.

Fine. I will try to get my hands dirty on that later on for the situation I have in mind. :)

Sometimes that's the only way forward. :)

Yes, I think that's what Peter May mentioned about the bad behaviour of the colimits

Yeah, that's a general problem when working with internal categories.

Did you understand his reasoning why the Nerve functor N:Cat(Top)->sSpace preserves the colimit in your situation?

Some regularity condition on the base category will ensure you at least finite colimits, but Top does not satisfy this criterion.

It seems the reason can be boiled down to the fact that closed injective functors are taken to levelwise cofibrations, and the union of any finite subsequence is respected by this assignment.

Colimits of simplicial spaces are very explicit since it's just a a functor category, so the only thing to check seems to be wether N_k(colim(C_i)) is homeorphic to colim(N_k(C_i)) as topological space

As before I suppose one gets a bijective continuous map immediately and the conditions should somehow ensure that the map is a hoemorphism

Yes, I think it is actually enough to check this for $N_2$, since you have the Segal condition.

Oh, I do not know about that.

Do all nerves satisfy this condition or is there another reason why it is fulfilled in our situation?

No, all nerves of categories satisfy this.

The Segal condition is just that $N_k$ is a $k$-fold pullback of $N_1$ over $k-1$ copies of $N_0$.

Even in the topological case without any regularity? I vaguely remember this to be true for discrete categories and simplicial sets.

Yes, this is another thing that hold for internal categories over any base, by definition.

Here: ncatlab.org/nlab/show/internal+category#internal_nerve

Nice. Thank you.

I mixed up things, it's for sure totally obvious

Happens to me all the time too. :)

Being there the situation gets nicer since the geometric realization SSpace->Space has a right adjoint, even in the topological case. The paper for that is somewhere linked on the nlab

I think this is the first point where one has to careful in which category of spaces one works. The paper proving that adjoint surely lives in some "convenient category" and I suppose that's really necessary. Cartesian closeness will enter the stage at some point, I think.

Can you tell me the title of the paper?

matwbn.icm.edu.pl/ksiazki/fm/fm106/fm106115.pdf §3

Thanks.

He works in CGWH, as I guessed

I would expect as much. Nobody really works with arbitrary spaces. :P

Yeah, but most of the people do also not think much about these subtleties. :D

Colimits in CGWH could be nasty, I guess

So if some of my (or your) arguments are proven by hand like the homeomorphism properties of continuous bijections, we have to be careful.

In general, yes. You do have some criteria for existence. The ones I've seen involve some cofribrancy requirement, like a pushout in which one leg is a closed inclusion, or a sequential colimit of cofribrations (as in my case).

For a sequential colimit of cofibrations of CGWH-spaces, the colimit in the general category of spaces is again CGWH and therefore the colimit in CGWH?

That should work for me, too if that's true

I suppose you can say that, yeah. You can also quotient out by closed equivalence relations.

The only real source of information about CGWH I have in mind is a survey of Neil Strickland, but that does not seem to address this case

Emily Riehl mentions it on page 80 of "Categorical Homotopy Theory".

Thanks, I'll have a look

Going from the colimit of categories to the colimit of nerves could make problems. I don't know whether nerve functor preserves cofibrations, so maybe it is not enough to prove that the nerve of the colimit category is the colimit of nerves with the usual colimits in all spaces.

Well, it is obvious on $N_0$ and $N_1$, right? Then you can use the Segal condition to show that you have a degreewise cofibration.

Colimits of pullback squares consisting of cofibrations are cofibrations?

Forget what I said

May I ask what you are writing your master's on?

My topological categories consist of cobordisms and I would guess yours do too. Am I right? ;)

Yes, they do. :)

So then I would guess you are also interested in Madsen-Weiss-theory?

Touché!

Cool!

The GMTW-Thm. is great. :)

Yes, but I have great difficulty wrapping my head around their sheaf methods.

Yeah, but there are ways to prove it without those techniques. I also like the scanning methods of Galatius.

The one that uses "microflexible sheaves"? I haven't looked closely at his paper yet, only the last section.

Using Gromov's h principle is one possibility, yes

But I don't know much about this either. I planned to skim through Gromov's book to get an idea of.

I kind of dislike it when authors write "the above methods work equally well for this unrelated problem" without going into details.

You mean the last paragraph of the paper which calculates the stable homology of the automorphism group of free groups?

The first sentence of the last section, yes.

But I guess he does give some details.

Are you coincidentally in Bonn next week? There are some lecture series for grad students in the realm of their current trimester program about homotopy theory, manifolds and field theories.

No, I won't be there. I heard about the trimester program, but it's inconvenient for me. I have to hand in my thesis next month, so I can't afford to spend too much time on other stuff. :\

I hope they put out lecture notes or something afterwards though.

I will at least take handwritten notes. I can scan you those, if they won't put some online, if you wish

That would be very nice of you!

I'd really appreciate that, if it's not too much trouble.

You're welcome. No problem at all

But I will excuse myself for my handwriting in advance :D

Thanks!

Have you ever seen Grothendieck's handwriting? I don't think it is possible to top that. :P

Oh, I think you even have more luck: Carmen Rovi is attending. She's a student of Ranicki, who records many talks on video. I can think of that she will do that next week, too. I'll keep you updated.

Great! Thanks!

Yes, I have seen that. It's awful

So you're already finished next month. :) Do you have plans on doing a PhD afterwards?

Yes! I have been accepted to Copenhagen University! :D

How about yourself? Do you have plans to continue after your master's?

Congratulations! May I know with whom you'll work?

Probably with Natalie Wahl.

I plan on doing a PhD, too, but there is nothing fixed yet.

Are you graduating next year?

By the end of this year, I suppose. But eventually I won't start doing doctoral studies right after. Nothing sure yet

That's cool. I skimmed through the websites of the university of Copenhagen, too. You have been accepted even before the application deadline? That's nice. :)

I see.

Which deadline are you looking at? The application deadline was in January.

Maybe I mixed up something. I'll take I look what I had in mind

math.ku.dk/english/about/jobs/phd-stipends-2015 That's what I saw

They seem to have two application periods a year

Oh, right.

I think that is standard practice actually. Bonn does that as well.

I have to restrain myself not to bomb you with questions right now, since I considered applying in Copenhagen, too. :D

Haha. Don't worry about it. :)

Should we change the place to chat? It's all public here

Yeah, ok.

Any suggestion?

Dunno. Facebook is always an option.

Okay, give me a minute, please

ok

Hmm, try this:

chatstep.com/#Espen%20and%20archipelago

the password is "mathoverflow"