Does anyone know if it is possible to prove the small object argument (or something similar) without using coproducts of maps?

I don't understand this very well but it seems that Emily-Verity have managed to develop a framework ($\infty$-cosmoi) in which the basics of $\infty$-category theory could be developed uniformly in a "classical" 2-categorical framework. Can the basics of $\infty$-Operads be treated this way? Is there an $\infty$-cosmoi of $\infty$-operads in which Operadic co/limits and kan extensions have 2-categorical universal properties?

I guess you could separate this question to two parts. Firstly, is it in principle possible to develop the theory of operads, algebras and their modules externally using only the $2$-category theory of operads without using anything about their internal structure (which I believe is true for categories). Secondly do we currently have the machinary to do so for $\infty$-operads?

@User1236262625 I think it should be possible, but it becomes a "scheduling" problem -- you've got to make sure you eventually add in all the lifts which make their way in there in the standard argument, and they all have to be there at once when you hit each sufficiently-large regular cardinal... In theory there's no reason to do this, though, because if your cofibrations are closed under cobase-change and transfinite compositition, then they are closed under coproducts.

The kind of issues you need to deal with appear when constructing Fraisse limits.

@SaalHardali Somebody asking Emily and Dom about this at Talbot last year, and what became clear is that there should be multiple ways to do it. Dom's favored approach is the "operads as analytic monads" approach. I don't recall Emily having so strong a preference, but she pointed out that the $\infty$-cosmos of $\infty$-categories should already provide a sufficiently rich setting to re-do Lurie's approach in a straightforward way.

Alternatively, one could build it from the ground up as you seem to be suggesting. If you wanted to use dendroidal sets, say, this might be the way to go.

But I think it's an enticing open problem of the "low-hanging fruit" variety.

@TimCampion Just to make sure I understand. They said they do believe that $\infty$-operads should form an $\infty$-cosmos and could be studied in this way? Nevermind the specific approach for a moment.

Well, maybe it was a bit too strong to say that precisely your question was asked -- I don't recall whether they discussed whether $\infty$-operads should form an $\infty$-cosmos. But sitting here, it seems they should. The only model-independent property an $(\infty,2)$-category needs to have to be an $\infty$-cosmos is to have finite 2-limits, which $\infty$-operads clearly should have.

But there are things to check, because there are certain model-dependent properties that an $\infty$-cosmos needs to have as well.

For instance, although there is an $\infty$-cosmos of quasicategories and an $\infty$-cosmos of complete Segal spaces (and they are suitably equivalent), there doesn't seem to be an $\infty$-cosmos of simplicial categories.

I see. So I can think of $\infty$-cosmos as modelling an ($\infty$,2)-category with finite 2-limits?

Yeah

And is there a non-taotological way of knowing what a 2-limit is?

I never understoon why Riel-Verity don't give more 'exotic' examples of ∞-cosmoi coming from weird (∞,2)-categories

I would really love if there was a way to do some basic stuff with operads using $\infty$-2 category theory. Like the operadic kan extensions

As far as the model-dependent properties, a useful sufficient condition is that (the fibrant objects of) any model category which is simplicially-enriched -- with respect to the Joyal model structure on simplicial sets -- is an $\infty$-cosmos.

@DenisNardin I'm with you! I think it has to do with the fact that the whole project is aimed at accomplishing several things at once. In particular, they're constantly trying to balance the opposing goals of providing a framework which (1) allows people to do very general things in a uniform way and (2) allows novices to learn $\infty$-categories much more easily in the first place

I guess given what Emily said knowing more about how to do the ($\infty$,1)-category in Lurie using $2$-category theory could help with doing the same for operads.

Yeah, I think you're right. For instance, one should really be able to say that an operadic Kan extension is just a Kan extension in the $(\infty,2)$-category of $\infty$-operads.

That's what I want to be true but it's not at all clear to me whether it is.

yeah

I suppose one thing to be careful about is that one of the featured aspects of the $\infty$-cosmos framework (at least as currently developed) is the systematic use of the notion of a cartesian fibration as defined internally in a 2-category.

What do you mean by "internally in a 2-category"?

If $p: E \to B$ is a 1-morphism in a 2-category, you say that $p$ is a cartesian fibration if for every $b: A \to B$, the induced map $p \times_B b \to p \downarrow b$ has a left adjoint left inverse.

where $p \downarrow b$ is the comma object, defined by a certain 2-categorical limit.

This definition works great for 2-categories like $Cat$.

But it turns out to not really be what you want in the $Ab$-enriched setting, say.

So you're saying maybe this definition doesn't give the right thing for operads?

Yeah, it might not. It should be straightforward to check whether it does for ordinary operads, though...

In Lurie a functor is a cartesian fibration if is on the category of operators. So maybe all one should check is whether this functor (sending an operad to its category of operators) preserves the property of being cartesian.

I won't feel like I understand what's going on until I understand what the "straightened" version of a cartesian fibration of $\infty$-operads is, and compare that to the 2-categorical context...

But as I think about it, one thing seems clear: Lurie's approach to $\infty$-operads is specifically designed to take advantage of all the $\infty$-categorical machinery built up in HTT

I discussed it a bit with Yonatan Harpaz in the past, and I think that the internal notion of cocartesian fibration gets the right one for operads. We never quite worked out all the details too

Isnt that just an algebra in $Cat_{\infty}$?

(This machinery is mostly available in the $\infty$-cosmos framework, so basically mimicking Lurie and doing $\infty$-operads on top of an $\infty$-cosmos layer just as Lurie does it on a quasicategorical layer might not be a bad way to go.)

@DenisNardin The"right one" meaning that the 2-categorical definition gives the notion of cartesian fibration of $\infty$-operads as in HA?

Is a cartesian fibration of operads over $O^\otimes$ the "unstraightening" of a morphism of operads $O^\otimes \to Cat^\times$? Or $O^\otimes \to Opd^\otimes$? Or something like that?

@SaalHardali Yeah, but it's cocartesian, not cartesian. It's very important because the (∞,2)-category of ∞-operad doesn't have an automorphism exchanging cocartesian with cartesian

@TimCampion It's the "correct" notion of $O^⊗$-monoidal category

Essentially it is the unstraightening of a functor $O^⊗→Cat$ satisfying the Segal conditions

Oh that's an obvious distinction I've never considered before. Thanks for pointing that out.

It's the same reason why there are left operadic Kan extension, but not right operadic Kan extensions

Ah, right -- so you're saying a _co_cartesian morphism of operads over $O^\otimes$ is a kind of functor $O^\otimes \to Cat$.

That does sound like the kind of thing one should be getting from the internal 2-categorical notion.

I suppose the other thing is that for a number of $\infty$-cosmos purposes, you restrict to groupoidal (co)cartesian fibrations -- basically requiring the fibers to be $\infty$-groupoids in some sense. Presumably one would hope that in the operadic case, a groupoidal cocartesian fibration corresponds to some kind of functor $O^\otimes \to Top$...

I suppose I'd like to add that what Denis is talking about appears to give you a good notion of $O^\otimes$-monoidal $\infty$-categories, but not $\infty$-operads.

@JonathanBeardsley Well, roughly in the same way in which cocartesian fibrations over C do not model all of Cat_{/C}.

Since $O^\otimes$-monoidal $\infty$-categories are just, as was mentioned, certain types of cocartesian fibrations over the category of operators of $O$. And I guess one could embed any reasonable $\infty$-operad into one's favorite $\infty$-cosmos (again thinking of it as a category of operators), but it'd be nice to have a description of $\infty$-operads within any $\infty$-cosmos entirely in terms of certain kinds of fibrations to $N(Fin_*)$ I think.

Something that I didn't understand very well when I first started reading this stuff was that $\infty$-operads, as functors to $N(Fin_*)$, are very much not cocartesian fibrations.

Anyway, that's not a particularly substantive comment, it's just something that I had to clarify for myself a while ago.

I think the analogy with cocartesian fibrations vs entire slice category is good only that the situation with operads is somehow worse. The cocartesian fibrations at least form a faithful $\infty$-subcategory in the sense that you choose some of the objects and some connected components of the morphism spaces. In the case of $\infty$-operads considered as categories over $Fin_{\ast}$ i'm not sure you can say the same... but I may be wrong.

@SaalHardali I don't know if fibrations of ∞-operads model the slice category. I suspect not, but I only wanted it to be a loose analogy anyway

@DenisNardin This is actually something I thought was true. Shouldn't fibrations over $\mathcal{O}$ model $Opd_{\infty / \mathcal{O}}$?

What I wasn't sure about is $CartFib_{/ \mathcal{O}}$ being faithful subcategory of the slice.

@SaalHardali Hrmm, what's unclear to me is whether the fibrations with a fibrant base on the model structure on ∞-preoperads are the fibrations of ∞-operads. But maybe that's true

@SaalHardali That is true, it's not too hard but it's buried in a forgotten corner of HTT that I am too lazy to hunt for right now

The trick is to use that the model structure on marked simplicial sets is simplicial (and so is the structure induced on the slice over a fibrant object)

@DenisNardin You mean my first statement is true right? or both?

@SaalHardali I was talking about the second statement

Oh, interesting. So you're saying the fibrations don't necessarily model the slice?

@SaalHardali Don't take me too seriously. I don't remember the details of the model structure, and a cursory search hasn't enlightened me

Sorry, just trying to understand where we stand about this.

So being an $\infty$-operad is really a property once you restrict to the subcategory of functors which are cartesian over the inerts. That's nice to know. I wasn't sure about that.

Hi all! New here.

Does anyone have a good reference for the result $H^*(u(n); F_p) \cong E_{F_p}(x_1,x_3,...,x_{2n-1})$, i.e., that the cohomology of the Lie algebra $u(n)$ -- the Lie algebra corresponding to the Lie group $U(n)$ of $n \times n$ unitary matrices -- is isomorphic to the cohomology of the exterior algebra over $F_p$ on odd degree generators ?

I believe the desired isomorphism is obtained by carrying out either the Hochschild-Serre spectral sequence, or the Serre spectral sequence. Does anyone know of a reference where they explicitly carry out this calculation? Thanks in advance.