anyone know whether the formalism of the cotangent complex has been worked out in the setting of (suitable) algebraic theories, instead of operads? is that a reasonable question?

@bananastack I decided to ask a question on mO.

@Adeel The only thing that rings a bell is Schwede's early work. He might have done this there. There is also, once some homotopy theory in placed, a notion that the cotangent complex should be some stabilization functor, but you probably knew this.

Right. I've only seen it worked out for infinity-operads though. Actually, what exactly is the relation between algebraic theories and operads? Are operads strictly more general? I think all the interesting infinity-operads I know can also be described by infinity-algebraic theories, while the latter seem so much simpler in my opinion.

ordinary operads (i.e. monochromatic, over Set) are strictly generalised by algebraic theories

That's interesting

it's obvious, though

Hello

@Adeel I can't seem to find a reference for the fact we discussed yesterday

Which fact exactly?

mathoverflow.net/questions/201651/… @Adeel

I can't find it in stacks, for instance

man, math.se has really degenerated

quality used to be a lot higher

I don't know a reference either. If it's not in the stacks project, I would try EGA IV I guess.

I'm tempted to send johan an email

I looked in milne and the only thing there is openness on the source

I actually misread your question yesterday apparently. I thought you were talking about openness on the source too. For the target I don't really know of such a statement at all

I think the argument in my question is ok. I just feel there's too many hypotheses

Like ideally it would be good to remove like either finite or loc. finite presentation

doesn't the definition of étale include lfp?

ok word

@Adeel I'm not completely sure, but I think that if you take the Goodwillie derivatives of a finitary monad you should end up with an operad and this should give you an adjoint to the functor that given an operad produces a finitary monad. At least in spectra this should work

So operads should be though of as "polynomial approximations" of monads

thinking about algebraic theories in terms of finitary monads makes it much harder to see the similarities and differences between algebraic theories and operads

@DenisNardin that went over my ahead i'm afraid

@Adeel The $n$-th derivative of a functor (at least in spaces or spectra, I'm not 100% sure how general this is) is a naive $\Sigma_n$-spectrum, so from every functor you get a symmetric sequence of spectra. When your functor has a monad structure this combines to give an operad structure to your symmetric sequence

@ZhenLin Possibly, I was just relating a cool fact about their relationship :)

can you calculate any examples?

so what are the advantages of operads over algebraic theories? it seems a lot easier to work with algebras over theories (they are just presheaves after all) than with operads, where you have to deal with (co)cartesian fibrations and other stuff I don't understand

The canonical example of this is the identity comonad in spaces (sorry, just had to pop out a dualization), where you get the commutative cooperad if I recall correctly

for example, wouldn't Higher Algebra be a lot shorter if it was written in the language of algebraic theories? or are there some concrete advantages to working with operads there?

@Adeel In higher algebra you want coloured operads to model symmetric monoidal structures. I don't know how to do that for algebraic theories

@DenisNardin that's because you use the wrong definition of algebraic theory :p

(btw I hereby propose that we stop talking about $\infty$-operads and we start calling them $\infty$-multicategories)

well, then an algebraic theory is a cartesian representable multicategory!

@ZhenLin I don't know, is this actually giving you a significant simplification of the theory?

But our multicategories are not always cartesian representable, that's the point

We want to deal with much more general monoidal products (e.g. tensor product in vector spaces, smash products in spectra)

of course, and that's why people study operads in the first place

would symmetric monoidal infinity-categories just be infinity-algebras over the (2,1)-algebraic theory of finite spans or something (the algebraic theory modelling E_infinity-rings i mean), in the infinity-category of infinity-categories, or something along those lines?

Uh that went over my head, but a symmetric monoidal infinity-category is just a commutative monoid in Cat_\infty

If that's helping

(seriously cocartesian fibrations are just an handy way to talk about functors to Cat_\infty, people should not be afraid!)

I think E_infinity-monoids in any infinity-category are just algebras over a certain (2,1)-algebraic theory

so here's the thing: in Set, at least, and surely also in ∞Grpd, every operad can be completed to an algebraic theory

cool

in particular, the operad Comm can be completed to an algebraic theory

plus, algebraic theories allow you to talk about strictly commutative monoids in infinity-categories

for example, the infinity-algebras over the ordinary theory of commutative rings are simplicial commutative rings

that's the only description of the infinity-category of simplicial commutative rings I know that doesn't pass through model categories (or relative categories)

Ok I have to ask this: what are strictly commutative monoids? What's the intuition for them?

Like, there's a forgetful functor to commutative monoids. Is it monadic? Comonadic?

i don't really know, just they are convenient for doing algebraic geometry

if there's any justice in the world, it will be monadic

I honestly don't know what to think about them. I don't have any intuition whatsoever. Are they a full subcategory of commutative monoids? (presumably not, but I don't know)

What does strict commutativity buys you?

I have no intuition either

but there's the closely related situation of simplicial abelian groups vs infinite loop spaces

Yes, and I don't understand that either! It's frustrating :)

i mean, geometrically, it means that the affine line is well-behaved

It would be nice if someone could exhibit an explicit equation that strictly commutative monoids satisfy that E_\infty-spaces don't

in geometry over E-infinity ring spectra, you have two affine lines... Lurie has made a comment on MO to the effect that geometry over simplicial commutative rings is what you get from geometry over E-infinity ring spectra by forcing the two different affine lines to coincide

@Adeel Well, yes, that's just a tautology isn't it?

depending on how you are making that a precise statement, I guess

I assume it's going to involve a coherent system of factorizations of $X\times B\Sigma_n\to (X^n)_{h\Sigma n} \to X$ through $X$ itself, but I don't know :) I wish someone told me exactly what this means

Anyway sorry for the rant, I'm going back to work

@DenisNardin btw, I just remembered that Lurie makes some precise statements about the forgetful functor {simplicial commutative rings} -> {connective HZ-algebras} in section 2.6 of his thesis

it commutes with colimits and limits, and {simplicial commutative rings} is the category of coalgebras over the comonad given by the adjunction (with the right adjoint)

Great :) Now we just need to describe the comonad (which probably will involve power operations in some form)

if the functor is plethystic, which I think it is, then the comonad is given by "E_oo maps from Sigma^\infty N"

that's not very explicit, though

@Adeel Something tells me user74230 is brian conrad

?

pretty damn sure, his comments have referred to papers by brian conrad, user seems very familiar with EGA and algebraic groups

doesn't he participate as BCnrd?

well bcnrd hasn't been active for a while

@Saul Yes, I would like to have a better description than that :) That's what I was referring to when I said that "collapsing the two affine lines" is a tautological description (you probably mean $\Sigma^\infty_+ N$, aka $\mathbb{S}[N]$ or the second affine line)

@SeanTilson there are actually multiple E_infty orientations of KU, but I think only one that induces the multiplicative formal group law (and it's a lift of the Todd genus). I don't know off-hand of a proof that the Todd genus is E_infty before Barry Walker's

@TylerLawson I would have that that it was classical by some kind of Atiyah-Bott-Shapiro type thing. Or maybe they didn't show that this was $E_{\infty}$ either?

@Sean I'm quite sure that ABS didn't show that their orientation is E_\infty and I don't know of any reference proving it either (apart maybe by applying the same proof as the string orientation of Tmf)

But I will be extremely happy to be proven wrong

It looks like I might be wrong and the reference is to Joachim

"Higher coherences for equivariant K-theory"

hmm, I never looked at that bu I own the book. Ha!

however, Joachim does the spin^c orientation. Atiyah-Bott-Shapiro predates the definition of being E_infty, but it is possible that it came earlier. hang on.

Thanks!

@SeanTilson I can't find this result in May-Quinn-Ray which would have been the main classical candidate.

Joachim says in the article you mention that this is the first proof in the literature even for the trivial group.

There you go.

In the end, I would want to know the effect on $\pi_*$ so it is unclear if that will be in here. Someone tried helping me trying to unwind the map from the Todd genus, but we couldn't do it.

The things in question are torsion-free and so you can probably just check what happens after rationalization.

(which probably amounts to doing some manipulations with Hirzebruch's characteristic series)

@BenLim user74230 is definitely BCnrd. The real question is: who is abx??? He/she knows everything.

who is bananastack is the real question

...darn

@bananastack You sure?

man this abx even knows the precise ega reference!!

who has the last initial X though?

what impresses me is that he knows all the foundational EGA stuff, all the birational geometry stuff, all the classical stuff (he asked a question with the word tritangent in it...) and once even answered a question on hochschild cohomology!

Who is @bananastack? Are you a grad student?

Can't believe brian conrad asked me to ask the experts at my undergrad institution. I just left the place!!

@Adeel if you figure out anything about the cotangent complex for algebraic theories I'd be very interested in chatting with you about it.

Does anybody know if Goodwillie calculus commutes with geometric realizations? I have a simplicial functor and know how to compute the derivatives levelwise, dreaming that it might be enough.

whoops, P_n is a left adjoint. Sorry for the dumb question.

@ThelSeraphim not sure whether it would be useful to work it out though. the greater generality is probably not useful to anyone, and it seems that everyone other than myself already understands infinity-operads anyway

Is it true that if I have two CW-spectra E and F that I can define E \smash F as the spectra which in degree n is hocolim_{i,j} \Sigma^{n-i-j} (E_i \smash F_j) , where i and j ranges over the integers and I take the hocolim in the Quillen model structure of spaces. Of course, I can define it that way, but does it agree with the old-fashioned way of defining the smash product of spectra ?

@Adeel it would be very useful to me, perhaps we could talk in a less crowded room: thel.seraphim@gmail.com

Hey, what's the spectrum associated to the sequence of spaces BU, SU, BU, SU,...? Is it still just KU? Or some version of KU?

@JonBeardsley Still KU. The natural map from it to the spectrum for KU is a stable equivalence.

Oh okay, thanks.

@Dedalus unless I'm mistaken, that's very similar to the approach taken by Adams in his "Stable homotopy and generalized homology". However, you have to be careful (you need to know that you have structure maps, which is not as immediate from your definition as you might like) and you'd be advised to not do it this way

@Tyler Lawson: do you have any recommendation / resource for a better way?

@Dedalus one of: (1) use a handicrafted smash product as Adams describes if you just need to know the homotopy type; (2) take it on faith that there's a good definition with good associativity properties or commutativity properties in a slightly different category; or (3) move into a category like symmetric spectra (Hovey-Shipley-Smith) or EKMM spectra or if you really need an in-depth definition. (there is also a version in Lurie's Higher Algebra, but you need to be in that "zone" already)

Stefan Schwede has a book project on symmetric spectra that people seem to be using as a base reference these days.

@Dedalus One I'm fond of is "identify spectra with reduced excisive functors from finite CW-complexes to spaces and then the smash product is a localization of Day convolution". There's no precise reference for this as far as I know though (and in particular it is NOT in Higher Algebra, whose definition is much more complicated)

Does anybody happen to have a pdf of "Fredholm Complexes" by Segal?

look into your email :)

Thanks!