Given a grouplike topological monoid M, I believe it's classically known that M is equivalent to a topological group. Does the same thing hold for grouplike Kan complexes which are monoids?

sure, the mooned structure is an particular kind of A-infinity structure, so being group-like it is w.e. to a loop space, and all (simplcial) loop spaces are w.e. to simplicial groups (Kan)

mooned = monoid + corrector

Haha, I like the mooned structure.

XD

1:40am here, makes sense

I think we really need to take advantage of using the monospaced math font...

?

For objects in a quasicategory I've been using the regular math font as in $X\in C$, but then if I want to pick some specific space in a model category which is equivalent, I write $\mathtt{X}$.

So, in particular, I've been writing $\mathtt{BGL_1(R)}$ a lot.

It really stands out, and we don't really use it for anything else.

$\mathtt{f:X\to BGL_1(R)}$

And I don't have to pick new letters, or put tildes all over the place.

No? Not feeling it?

Plus I feel like I'm Larry Smith writing a paper in the 70's or something.

It's really distracting

Haha.

also, as to the first point, what do you do in writing?

also, really ugly

Like, writing with my hands?

yeah

I dunno, I don't talk to anyone but myself.

And so I know what I mean.

Or, I guess, I just don't differentiate. I just tell whomever I'm talking to which one I mean.

But I'm writing something where I kind of go back and forth, so don't want to have to keep explaining "and here we mean X in the quasicategory."

I suppose it could also cause serious problems if I try to like, send this to an actual journal, haha.

it's also hard to read

I'm going through and changing a bunch of things to it... not too pleasing.

I guess I'll just use \overline or some junk.

:(

although LaTeX doesn't do a great job with overlines either.

They should fix overline so it doesn't hang over the edge of the serifs. I shouldn't be as wide as the base of the letter, but only as wide as the top. E.g. $\overline{X}$

Well, it looks like garbage on here anyway.

$\overline{A}$ though?

Ah.... fair point.

Also not sure how one would handle $\overline{O}$ haha.

Ugh, literally everything I'm trying: \tilde, \widetilde, \overline, \mathtt looks like garbage.

Hah, speak of the devil... this person is using \mathtt arxiv.org/abs/1403.4130

I personally find it cute, actually

Here's an annoying question: what's an example of a homology theory on pointed CW complexes which satisfies Milnor's additivity axiom but not the direct limit axiom? (i.e. the homology of an infinite CW complex is the colim of the homologies over the finite subcomplexes) Any homology theory given by a spectrum will satisfy this stronger axiom

@TylerLawson I'm a bit confused. W-spaces = functors from pointed finite CW complexes to pointed spaces, which are topologically enriched and based. This has a model structure under which it is Quillen equivalent to e.g. symmetric spectra, and it has a good smash product. I don't see where the 1-excisive condition comes in.

oh, apparently this would guarantee that the associated prespectrum to a W-space is an omega-spectrum, and it's more or less equivalent to it.

@:2573748 I'm guessing the "prolongation functor" from prespectra to W-spaces is E maps to $\Omega^\infty(E \wedge -)$? Then I guess it's nice to have E be an Omega-spectrum, so that when we apply homotopy to this thing we will get the honest homology groups (without having to take a fibrant replacement). Is this why @DenisNardin put the "1-excisive" condition in his definition?

that last comment is not very correct...

wow, so apparently the direct limit axiom needed for Adams' representability theorem actually seems to be equivalent to the usual additivity axiom? mathoverflow.net/questions/229612/…

does anyone have Goodwillie's Calculus I & II papers and would kindly send them to me? Apparently my university doesn't have access to them

email: [email protected]

emailed

those papers are hard to find because they appeared in the now defunct journal K-theory

@SaulGlasman thanks!

if you want to learn calculus I would recommend looking at II and III, I really just contains a particular application to K-theory of spaces iirc

specifically, it should be possible to access the archive through Portico

ah, that's where mathscinet redirected me, but apparently my uni does not pay Portico

@lenticcatachresis i thought this would tell me who you are... but it remains unclear

Google just keeps asking me if I want to know about fried gherkins.

Which obviously I do, just not right now.

It should have suggested Freyd gherkins. by now it sure knows you're keen on category theory

haha

Hi @UrsSchreiber! don't see you in here much

@JonBeardsley i do exactly this -- throughout the project i just posted, there are just a small handful of places where i need to choose a specific quasicategory that presents an $\infty$-category (or similarly)

@AaronMazel-Gee Hah, funny.

Yeah there's only one place that I need to do it at present, and it's so jarring that I decided to not do it.

Hey - question for the room: If I want a mapping space to be "derived," I need it to be from a cofibrant object to a fibrant object, Hom(X,Y) where X is cofibrant and Y is fibrant. This is supposed to make it "homotopy invariant" right?

yes, exactly. (i assume you're in a simplicial model category now?)

Yeah I suppose so.

if you change the source by a weak equivalence to another cofibrant and/or the target by a weak equivalence to another fibrant, you get a weak equivalence of kan complexes (which is then actually a homotopy equivalence)

So I guess, and I know this is stupid, so take it easy on me here... something's confusing me. If it's now homotopy invariant...

Okay right exactly.

But let's say we've got Hom(X,Y), X is cofibrant, and Z is equivalent to X, but not cofibrant.

Ok nevermind.

So you can only change to other fibrant or cofibrant objects.

@AaronMazel-Gee but it's that left pointing iso i don't understand...

That's saying that if X is cofibrant and X\simeq Z, and Z is not cofibrant, then Hom(X,Y) is equivalent to Hom(Z,Y), right?

are you sure you mean to be replacing Y with Z, and not X with Z?

Whoops.

(also, the leftwards map isn't an iso, it's a weak equivalence of relative categories)

Hm okay.

I guess I'm saying that if Hom(X,Y) is equivalent to Hom(Z,Y), doesn't that mean that Hom(Z,Y) is also homotopy invariant?

is Y a fixed variable here?

or i guess you mean "homotopy invariant as Y varies"?

Y if fibrant and fixed.

Is Hom(-,Y) homotopy invariant?

okay, so then what does it even mean to be "homotopy invariant"? in general, this is the request that weak equivalences are taken to weak equivalences

for Y fibrant, hom(-,Y) is only homotopy-invariant when restricted to cofibrant objects (and weak equivalences among them). it might happen that its value on a non-cofibrant object Z agrees with its value on a cofibrant replacement X of Z, but that'd just be a coincidence

I see. Okay.

And similarly, presumably, for cofibrant X (restricted to fibrant targets).

yes, the theory of model categories is entirely self-dual

rite

By a paper of Hewett we know which finite group can be a subgroup of our Morava stabilizer groups. But do we know how to write generators of those subgroup down explicitly? Hewett's paper makes me a little headache