Homotopy Theory

12:03 AM

Ugh. What the hell is a semi-inert morphism?

Oh I see. The inverse images have at most one object.

Instead of exactly one object.

Well now that I know that, the $\infty$-operadic definition of module categories is completely transparent⸮

@ZhenLin do you happen to have a reference for this fact from Lurie? I guess the above cited Joyal reference is fine, I just feel like I've read this in HTT at some point.

Zhen Lin

oh, I'm sure it's there too

but the place I remember seeing it is the early Joyal paper (2002)

hmmm... possibly wrongly, because I can't find it in there now

Jon Beardsley

Hey y'all, how does this stuff Lurie writes about on Module categories relate to this $\mathcal{BM}^\otimes$ operad he constructs?

Or, maybe he says this at some point.

Zhen Lin

I guess the statement that a natural transformation is invertible if and only if it is componentwise invertible is a special case of some fact about marked simplicial sets; at least that's how Riehl and Verity prove it in [The 2-category theory of quasi-categories]

hmmm... yes, so it's a special case of Proposition 3.1.2.1 in [HTT]

Josh Shadlen

3:11 AM

Did anybody here ever get a good handle on what 2-equivariance means? There is a kind of appealing sketch in Lurie's Elliptic Cohomology paper but I don't see how to state what a 2-equivariant cohomology theory is abstractly, or any desiderata restricting what it assigns to K(Z,3) etc.

For ordinary equivariant cohomology theories the main thing that one might not guess is the existence of Thom isomorphisms for equivariant vector bundles. Is there an analogous geometric operation that becomes available in the 2-equivariant world? I think it can't be as simple as inverting BS^1-representatio…

Peter Nelson

4:17 AM

related?: do the words "2-representation sphere" make any sort of sense?

Jon Beardsley

@JoshShadlen i haven't heard of this at all, but your comments make me think this is supposed to be something about equivariance with respect to a delooping of a group or something?

Josh Shadlen

5:02 AM

@PeterNelson one probably bad definition would be a vector space V and a pointed map K(Z,3) to BGL(V) -- then take the one point compactification, but I think it might not be hard to prove they're all trivial. I doubt this is the right way to think about it.

@JonBeardsley yeah something like that but it's pretty mysterious! I'm asking basically what "genuine" 2-equivariant cohomology looks like.

Jon Beardsley

So, is this under the assumption then that the group of interest is not commutative?

Peter Nelson

I would certainly like to assume G is non-commutative

(In particular, I like symmetric groups)

Jon Beardsley

I mean, otherwise its delooping is still a group, and so we've got a notion of genuine equivariant cohomology. Right?

Peter Nelson

I guess, selfishly maybe, I have a notion of what a 2-representation is (arxiv.org/pdf/math/0602510v4.pdf) and want to know if, maybe, one can index spectra by these sort of things...

Jon Beardsley

I guess, just the point is to do equivariant homotopy theory w/r/t a 2-group instead of a group?

Qiaochu Yuan

9:57 AM

so, my impression is that the point of thinking about representation spheres is to get thom isomorphisms so you can get, say, poincare duality for G-equivariant manifolds or something like that. is that right?

peter may said something once about how in full generality you should actually index spectra by the picard group of the equivariant stable homotopy category, but that's too big so you just look at representation spheres because they're useful and tractable or something

so, ignoring entirely the subtleties around genuine equivariance, the most general thing we could index "genuine 2-equivariant spectra" by is the picard group of the "2-equivariant stable homotopy category," whatever that means

again ignoring sublteties around genuine equivariance, that should look something like spheres with an action of the 2-group, or more homotopically like spherical fibrations over the classifying space of the 2-group

the reason we can get something like this from vector spaces is due to the J-homomorphism BO -> BPic(S). now in the 2-equivariant story if we want to think about groups acting on 2-vector spaces then the replacement for BO is something like the connected component of K(ko), and then I guess we can try to look for a "2-J-homomorphism"

or i guess to go in the opposite direction, having a good answer to the question above about 2-representation spheres seems at least as hard as having an answer to the question of what a 2-J-homomorphism looks like

user101036

12:30 PM

If I have a category cofibered in groupoids , i.e a map p:C--> D this is the same as asking it to be a left fibration when we pass to the nerve. Now, what is the equivalent of asking p to be a kan fibration resp. a trivial kan fibration

?

or hm, between ordinary categories it is just something surjective on objects and an equivalence I suppose

but appearently, if the fibers of a left fibration p:X-> S are all contractible it is Kan. So i don't see why it should be true, for nerves of categories, but I think it stems from a lack of knowledge of categories cofibered in groupoids. So, if it is contractible , the functor is clearly surjective on objects since the fibers are non-empty. And contractible fiber means what exactly?

user101036

12:56 PM

is it just that it allows us to define a functor q:S--> X as follows: for every s \in S choose some x_s \in X_s that X_s contracts onto and set q(s) = x_s and for morphisms, we choose a lift of \Delta^0 \rightarrow \Delta^1 and so on. Of course, all this must really define a functor so there are things to check and probably fix, but is this the general idea?

Zhen Lin

1:21 PM

if you replace all but two of the lifting properties with a unique lifting property then you get exactly the functors that are simultaneously Grothendieck fibrations and opfibrations as well as being fibred in groupoids

precise statement is 1.11.6 in my notes

user101036

1:33 PM

I will look into your notes then, thanks. I think I have an argument for showing that if a category cofibered in groupoids is such that all fibers are contractible, we should be able to define an equivalence of categories. Does this sound reasonable to you?

Zhen Lin

2:06 PM

I think it should be possible to show more directly that such a functor is fully faithful and surjective on objects

user101036

Surjective on objects is straightforward, probably one could do something similar for fully faithful, but I don't see it now

Denis Nardin

Contractible for a category is equivalent to say that between any two objects there's exactly one map. This should be enough to prove the full faithfulness

Zhen Lin

2:23 PM

First, recall that a category is opfibred in groupoids if and only if every morphism is cocartesian. So pick a parallel pair of morphisms upstairs, and suppose they have the same image downstairs. Then you have two cocartesian lifts of the same morphism with the same codomain, so there must be a vertical automorphism relating them; but the only vertical automorphism is the identity, so the two morphisms we started with must be identical

user101036

Superb!

Qiaochu Yuan

8:45 PM

I don't really understand why people ask what prerequisites they need to start studying X before they've just tried to start studying X. if you start reading a book on X and it clearly uses stuff you haven't learned then you now know what you're missing!

Jon Beardsley

Hey @DenisNardin regarding our discussion about determining eqiuvalences of natural transformations objectwise, i think what i was really getting at is the following: coming from a model category, i might have a morphism of cosimplicial objects, $F:C^\bullet\to D^\bullet$. I know that it's morphism of functors of 1-categories. the issue seems to be, how do i know that it's a morphism of of functors of $\infty$-categories?

So the issue gets transferred from needing to check it's an equivalence on higher cells to needing to check that it's actually a well defined natural transformation of functors of $\infty$-categories.

But perhaps one can do something like start with a natural transformation on the level of model categories and apply the simplicial nerve to make sure that it carries over to higher morphisms.

@QiaochuYuan Meh. People are highly irrational creatures. =P

Adeel

9:34 PM

@JonBeardsley where is your morphism coming from?

Jon Beardsley

so, i'm in particular applying the thom isomorphism levelwise to the amitsur complex of $S\to MU$

Adeel

@PeterArndt hi!

Peter Arndt

hi!

Jon Beardsley

hey @PeterArndt! haven't seen you since the chromatic homotopy talbot!

Peter Arndt

man I've been caught spying in the homotopy chat room

Jon Beardsley

9:37 PM

lol

Peter Arndt

i guess now I've gotta explain myself :-)

Adeel

haha

Jon Beardsley

it's okay. we're familiar with lurkers around here.

Adeel

@JonBeardsley that whole sentence pretty much went over my head i'm afraid... but it sounds like you're working with cosimplicial spectra or something?

Jon Beardsley

oh yeah sorry.

i mean

Peter Arndt

9:43 PM

@Jon Well, good to meet you this way at least! And gladly the having to explain oneself got deflected on you, thanks to Adeel

Adeel

heh. how is the thesis going by the way?

Jon Beardsley

yeah so @adeel u know there's this thom iso $MU\wedge MU\to BU_+\wedge MU$?

Adeel

i know zero (0) things about spectra... but ok sure

Peter Arndt

I curse my thesis!! (which means that I'm at it...)

Adeel

hahah. i can understand the feeling

Jon Beardsley

9:45 PM

oooh, uh, sorry, haha.

but yeah, i've got a morphism of cosimplicial spectra in, say the model category of EKMM spectra

and i know it's an equivalence on objects

and i want to say that it's an equivalence of the associated diagram of cosimplicial spectra in the $\infty$-category of spectra

Adeel

no need to apologize. i need to learn this stuff someday

right

Jon Beardsley

@PeterArndt i'm doing well! i'm also writing a lot right now.

kind of stressed out with all of that

haha

Adeel

so it's a weak equivalence in your model category right?

it's then completely formal that it's an equivalence in the associated infinity-category

Jon Beardsley

Yeah.... I guess so. I mean... somehow what I'm getting at is not formal.

Dylan Wilson

@Adeel gotta be a little careful here, actually. He needs to make sure he's working in some model category of cosimplicial guys that presents the infty-category of cosimplicial guys.

Jon Beardsley

9:55 PM

I've got concrete spectra, Thom spectra, that have geometric constructions.

Adeel

i assumed that was the case

Jon Beardsley

And so I'm looking at their manifestations in two different models.

I feel like this is a sort of subtle point that people tend to brush under the rug.

I mean, not this specifically, but this kind of thing.

Dylan Wilson

also, to answer the question you asked Denis above, you'll need to be sure it's a morphism between fibrant-cofibrant objects in, say, the Reedy model structure. then it gives you a map in the underlying infty-category. (I mean, you'll always get a map, but it might not be what you think it is unless what I said holds)

J. Musser

I don't know anything about homotopy theory, but I do live in a nearby field...

Dylan Wilson

but really it's best, when building (co)simplicial guys, to just do it from the start at the level of infty-categories. there's this helpful lemma in the appendix of HTT on Reedy categories that says that, if you want to define a functor out of a reedy category and into an infty-category, you can do so inductively by sepcifying maps between latching and matching objects. the same goes for natural transformations between such things.

Jon Beardsley

10:00 PM

@DylanWilson Yeah. Agreed. Or at least, this is certainly believable to me.

Hi @J.Musser, you certainly don't need to know any homotopy theory to chat in here, haha.

@DylanWilson yeah, that's essentially what i'm in the process of trying to write down rigorously. that is, starting with the thom spectrum definitions in the second half of ABGHR and so forth

J. Musser

@JonBeardsley Haha! I was trying to get banned, for whatever reason.

Jon Beardsley

@J.Musser oh i see. alright. yeah i dunno, i guess you'd have to really start spamming the room.

didn't know people felt so strongly about this room...?

J. Musser

@JonBeardsley They were showing how dumb the MSE chatters (ie, me) are. ;)

Adeel

...what in the world

Jon Beardsley

@J.Musser just to be clear, this isn't the MO chat...

this is a specialized chat room for people doing homotopy theory, not just MO users.

Perhaps you're looking for:

MathOverflow

General discussion for mathoverflow.net

If you're interested in comparing "intelligence."

J. Musser

10:08 PM

@JonBeardsley No, I know which room this is.

:D

Jon Beardsley

Oh okay. Right on.

Adeel

i kind of wish it wasn't possible to lurk here without logging in

Jon Beardsley

Okay, here's a more general question about $\infty$-operads: we can presumably define $E_k$-algebras of a symmetric monoidal category $C^\otimes\to N(Fin_\ast)$ by morphisms of $\infty$-operads over $N(Fin_\ast)$, $E_k^\otimes\to C^\otimes$ right?

Aaron Royer

@JonBeardsley Yes.

Jon Beardsley

but on the other hand, Lurie defines associative algebras of $C$ by taking the induced planar operad $C^\otimes\times_{Fin_\ast} Ass^\otimes\to Ass^\otimes$ annd asking for sections of this.

are these equivalent constructions?

Aaron Royer

10:13 PM

@JonBeardsley Yes.

Jon Beardsley

in particular, can I get $E_k$-algebras by looking at $C^\otimes\times_{Fin_\ast}E_k^\otimes\to E_k^\otimes$?

and asking for sections

okay

Aaron Royer

All that pullback is doing is forgetting your symmetric monoidal structure to a monoidal one.

J. Musser

@Adeel I am logged in.

Jon Beardsley

Right.

Aaron Royer

This works for E_k, too

Jon Beardsley

10:14 PM

Okay. Good.

I prefer the pulling back and asking for sections construction.

CPM

it says somewhere on the nlab (I cant check because it is down) that the category of Schur functors (meaning "polynomial" functors F:Core(FinSet)---> Vect) is the free symmetric monoidal abelian category on one object, how do you see that (and what precisely does it mean?)

Qiaochu Yuan

10:46 PM

it means that if C is another symmetric monoidal k-linear (you need this if Vect means k-vector spaces) abelian category and S is Schur functors, then the category of symmetric monoidal k-linear functors S -> C is naturally isomorphic to C

*equivalent

here's an easier claim first: FinSet^{\times} itself is the free symmetric monoidal category on one object

(equipped with disjoint union)

here's a related claim second: the category of Set-valued presheaves on FinSet^{\times} is the free symmetric monoidally cocomplete category on one object (equipped with day convolution). this essentially follows from combining the first claim above with the universal property of the yoneda embedding

(another name for this category is the category of species)

a closer related claim: the category of Vect-valued presheaves on FinSet^{\times} (here Vect means all k-vector spaces), or "vector species," is the free symmetric monoidally cocomplete k-linear category on one object (equipped with day convolution). same deal as above but with the yoneda embedding for k-linear categories

weakening "cocomplete" to "finitely cocomplete" should get you some version of the result you want

(in particular you get a somewhat more general universal property)

(the point being that "polynomial" should be equivalent to "finite colimit of representables")

CPM

11:24 PM

Thanks for that

MathOverflow

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