what is the proper citation for joyal's foundational work on quasicategories? is it still HTT's [44] "Theory of quasi-categories I", which is still "in preparation"?

are all sset-enriched categories cofibrant? it kind of looks like it from the generating cofibrations at the top of page 4 (arxiv.org/pdf/math/0406507v2.pdf), but that doesn't feel like it should be true, and in fact i think i've even asked this same question here before

wrote up what I learned from that conversation above:

@Clark, I hope you don't mind me citing your comments in chat

@QiaochuYuan Not at all.

@AaronMazel-Gee I don't think so, but a counterexample isn't on the tip of my tongue.

Sometimes I think it might be worth not having a very good job if it means I don't have to actually write down very much.

I think I've learned a lot by writing things down

I don't think I'm learning much from what I'm currently writing. Maybe.

i feel like i always learn from the writing process, except for the ultra-annoying tail end of it when all my perfectionist tendencies drive me crazy

Wow, yeah, I dunno. Maybe I'm doing something wrong. I usually write things about half-way, then feel convinced that all my statements are true and then just have zero motivation to finish.

i think you should write them down way closer to 100%

the polishing phase can be a real pain, but the rest can be extremely useful and clarifying

@SaulGlasman hey do you know, in the McCarthy-Minasian paper, Lemma 5.7, what does $X$ have to do with the statement of that Lemma? Do they really mean for $X=S^1$ or something?

And then the more general statement should hold for something like a "blob-homology-etale" condition (for X-blob-homology)?

Here's a link to the paper, BTW: arxiv.org/pdf/math/0306243.pdf

I'm also pretty new to the "writing things maybe for public consumption" thing so I probably don't know anything. but I have learned that writing is hard

@JonBeardsley no, they mean the general thing. see equation (1) in the intro

also btw "blob homology" is already a different thing

@AaronMazel-Gee but in the statement of that lemma they say the map is thh-etale if and only if the etale descent formula holds

and then state the etale descent formula in terms of a general X

surely this is some kind of mistake

iirc saying it for S^1 implies it for all X? it's been a while since i looked at this

Okay fair enough.

Yeah perhaps that follows essentially from Proposition 5.2. It's not particularly explicit.

That is, they seem to indicate that thh-etaleness implies equivalence of those two towers regardless of the simplicial set that one is using.

An orientation induces a Thom isomorphism, does a Thom isomorphism induce an orientation?

I guess so right, since local trivializations are supposed to be the same data as lifts of the map defining the bundle.

Wow, that could be really confusing if you haven't been inside of my brain for the past 15 minutes.

I'm thinking of a map of (at least) loop spaces $f:X\to BGL_1(S)$ and then saying that $Mf$ is $R$-oriented when the composition (along the unit map) $X\to BGL_1(S)\to BGL_1(R)$ is null.

@ZhenLin do you know the answer (or recall the outcome of our previous discussion) to whether all sset-enriched categories are cofibrant? i actually checked your notes too, but it wasn't clear from there either. at first i thought we should easily be able to build up any object we want from the generating cofibrations (of your Definition 2.11.5), but then i realized that a pushout of categories is not computed locally (i.e. hom-sset by hom-sset).

...which is good, though, because otherwise my project would be a lot closer to trivial, since we can compute the localization $C[[W^{-1}]]$ of a relative $\infty$-category by taking the pushout $W^{\mbox{gpd}} \leftarrow W \rightarrow C$ (in $\infty$-categories)

does anyone have a favorite geometric consequence of BSp(1) not splitting off of BSp (as opposed to BO(1) and BU(1), which do split off of BO and BU)

i can manufacture consequences, but none of them rate as interesting, nevermind being anyone's favorite

@JonBeardsley This is basically the definition in ABGHR. More precisely, if that map is null then the associated Thom spectrum is orientable, and a nullhomotopy of it is an orientation.

@AaronMazel-Gee I don't remember the discussion, but no, not everything is cofibrant in the Bergner model structure

okay, cool. do you have an example off the top of your head? and also i'm guessing that in particular, not all fibrant objects are cofibrant?

I don't know for sure, but probably any non-free ordinary category will be non-cofibrant

your observation about localisation is precisely the justification that appears in Simplicial localizations of categories

although they only work in the model category of simplicial categories over a fixed object set

@AaronMazel-Gee here are some pretty extensive notes by Joyal math.uchicago.edu/~may/IMA/Joyal.pdf

@ZhenLin right, i was just being dense for a minute ;o)

@Adeel thanks, is this meant to address my question about sset-enriched categories?

No, just your previous question about Joyal's foundational notes

@Adeel haha right of course

it seems to be missing a lot of proofs, compared to these other notes of Joyal

yeah, nlab says: "For several years Joyal has been preparing a textbook on the subject. This still doesn’t quite exist, but..." and then they go on to cite these

@ZhenLin yes exactly

these two together

Yeah, Joyal is apparently super perfectionist about his writing

man, him and me both

i've been working since about 8 this morning

(it's now 1am)

i took a break for about an hour to have lunch and read about D-modules :o)

but other than that i've been writing

well, writing and fussing. lots of fussing.

this friggin project. i wrote a "terminology, notation, and conventions" bit that was supposed to be a subsection of the intro, but it ended up being 4 pages long.

this is what happens when you mix 4 different models of "\infty-categories" in the same project, i guess -- i basically copied the intro to barwick--schommer-pries to quell the anxious reader

well, maybe "quell the anxious reader's anxiety" would be a better usage of the word

anyways

quell (verb): put an end to (a rebellion or other disorder), typically by the use of force.

Haha. Thesis?

four...!

@Adeel yes, more or less. @ZhenLin yeah...so far. these are just the conventions for the first paper

i'm thinking i should probably keep the other stuff for later papers, e.g. i've got a whole thing written up about the grothendieck construction (i.e. straightening & unstraightening), but that doesn't get used til paper #2

btw, i'm more than happy to have y'all read and give me some feedback, if you're willing

@AaronMazel-Gee i'd certainly be interested in reading it, though I can't say how helpful my feedback will be

what is it about?

So, yea... please keep spam or offensive flags for stuff that's actually spam or offensive. I don't believe that message falls under either category. If you believe it's off-topic, tell the user directly or, if you really must, flag for mod attention. Spam or offensive flags have rather specific connotations, punishments and alert every single 10k user across the whole chat network.

@ZhenLin actually, it's right up your alley -- this sequence of papers is laying out the foundations of the theory of model \infty-categories. this first one sets up the "kan--quillen model structure" on the \infty-category of simplicial spaces. surprisingly (or maybe not, if i've told you about this before), this is cofibrantly generated by the same sets as $sSet_{KQ}$ (considered as levelwise-discrete sspaces). and as a bonus, it's also proper.

@Adeel awesome, thanks. i will send you a draft when it's ready

I'm not sure you told me all the details before, but you did mention a model structure induced by "geometric realisation" in the (\infty, 1)-sense

but please do send me a copy

@ZhenLin great, i'll do that

thank you @snailboat :-)

for some reason Morava reminds me of Gromov and Gromov reminds me of Morava

I'd love to hear them have a conversation

Tobi has a cool looking paper out this morning. arxiv.org/pdf/1411.1711v1.pdf

If I have a morphism p:X--> Y in a model category and some other map i:A--> B and a commutative diagram relating these morphisms, if I have a lift of i to X, under what conditions will lifts be equivalent up to homotopy?

Are you lifting i along some map?

right, hm, I wish I could draw a diagram

wait

upload.wikimedia.org/wikipedia/commons/thumb/9/91/… Behold

Haha.

I mean, when we can lift g to a map C--> B such that the diagram commutes, when will such lifts be unique up to homotopy?

Maybe I should start to use that sentence whenever I give a talk after I've drawn a commutative diagram. "Behold!"

Yeah I dunno, maybe one could construct some ad hoc subspace of morphisms and say something about it being contractible.

And yes, I think we really need to increase our usage of "Behold!" in homotopy theory.

@user101036 As you probably already know, this is true when g is a cofibration (resp., trivial cofibration) and h is a trivial fibration (resp., fibration).

right, I knew of those cases. Maybe those are all the cases one should expect when we are working with a general model category

Yeah, in that generality, I can't imagine a proof that doesn't use the conditions on both g and h.

this should be related to homotopy-orthogonality

I think there are some remarks about it in Hirschhorn §17.8

I'll look into it, thanks!

what's a good guideline for determining whether something should be a Lemma or a Proposition?

In my mind Lemmas are strictly for feeding into something else.

They are never leaves on the tree, always branches leading to leaves.

Propositions are small and not particularly exciting leaves.

That might become branches later in the paper, but not immediately.

Ok the tree analogy kind of broke down there.

Lemmas should be things you are comfortable omitting if you are giving a presentation without proofs

Also, I've never seen anyone write something like "We need to first prove the following technical proposition..."

It's always a technical lemma.

hah

@JonBeardsley I guess the analogy still works depending on what your relation is with leaves

Haha. Yeah .

I think it's quite clear that a properly structured mathematical document should be structured as a directed acyclic graph

I'm trying to remember a good math text that has one of those really complicated directed graphs in the front telling you the possible orders in which you can read the chapters.

Rudyak's Thom Spectra book has this, but it's very simple.

Leitfaden? SGA 4 has one

That would make a great question!

In my opinion, an ideally structured mathematical document should be more or less like a coq program, or package of coq programs, but in English

I dunno. I like stories.

i like to be able to come up with my own story

just looking at the definition of algebras over a Lawvere theory... are modules also encoded by this theory somehow? for example, simplicial commutative algebras are algebras over a certain Lawvere theory, can I say anything about modules over them, staying in this Lawvere framework?

Well, sometimes I feel as if the author knows the way to the story much better than I will looking at it for the first time (or even second time)

Serre's books come to mind. They somehow have just the right amount of storytelling

@Adeel A Lawvere theory could equally well be encoding "groups" or "rings", and the notion of an action isn't implicit (e.g. in the former case, does it act on a set or an abelian group?) You probably need a "colored" theory, encoding pairs of an object A^0, an object A^1 that it acts on, and all the possible operations \prod_k A^{i_k} -> A^j

@TylerLawson i see, thanks

@user101036 I agree with this. It helps me very much to have a sort of story/motivation before I start just reading abstract theorems.

Now The Stacks project contans some expository work if I recall correctly, but still, if one would strip away all the expository stuff there and just keep the theorems, I would think t would be a very very tough read

Like when first encountering schemes, it is nice to see drawings of them.

Wait, how do you draw a scheme????

Well, some affine schemes then

(I basically just imagine a 2D manifold with a sort of bundle of straws on top of it)

you have probably seen this before but: pbelmans.files.wordpress.com/2012/12/mumford-reprint-600.png A "picture" of Spec Z[x]

@JonBeardsley Eisenbud and Harris's book has lovely pictures.

@user101036 i haven't seen that. it's cool.

In general one way to think about affine schemes is like a parameterized family of varieties

(at least if it is of finite type)

For me I basically think of algebraic geometry as just parameterized algebra. =P

Since well you always have a map Spec A --> Spec Z, and you can look at the fibers

I think that is a nice way to think about it

But that's sort of backwash from thinking about parameterized homotopy theory.

That I know nothing about however

Getting good at drawing these schemes is really a great way to understand how the geometry encodes algebraic and number-theoretic information. You could spend many many hours just drawing various curves over Spec Z and understanding how their fibers look over various primes.

right! It is fantastically beautiful really

Not really related, but I spent some time a while back drawing the Hochster duals of varieties whose topology was easy to understand. That's kind of fun.

@Adeel I think it might be possible to adapt the definition of modules for an algebra for an operad to the Lawvere theory case, but I never got around to working out examples

the nLab page ncatlab.org/nlab/show/algebraic+theory says: "A ring A determines and is determined by an algebraic theory, whose models are left A-modules..."

sure, there is a Lawvere theory whose models are A-modules, of course.

maybe this is an indicator that modules over a T-algebra are also models for a Lawvere theory, for some Lawvere theory T?

well i guess for simplicial algebras, it's enough to pass to infinity-algebraic theories

According to Proposition 7.29 of ABGHR, we can think of MU as the quotient of S by the infinite unitary group.... wut?

I mean, I guess that's sort of obvious. But... what does this mean geometrically?

but in the first place, what is a module over a T-algebra?

right... never mind, just being stupid again

I bet there's some clever way to define modules over algebras for a general Lawvere theory

there is a definition for operads, which could be extended to Lawvere theories

if I work over the category of G-simplicial sets , G a discrete group, with model structure such that f:X --> Y is a w.e. iff it is so for all normal subgroups, will then weakly equivalent spaces have weakly equivalent orbit spaces? It seems so, but maybe I am wrong here

since we should have a quillen adjunction to simplicial sets where the left adjoint is colimit, so it should be homotopical there

@JonBeardsley What do you mean by "geometrically" here?

I tend to interpret that Proposition as saying that Thom spectra are the "\infty version" of group homology.

That's not terribly geometric, though.