@MarcHoyois Oh. That's way easier. Thanks!

@SaalHardali The reason why the internal operads seem like the right idea in a topos is that since the internal logic of your spaces treats objects of T as sets or spaces, there's no reason why you wouldn't want to relativize the object set. But yeah if you want T-enriched operads, they shouldn't be significantly different in a topos vs any other symm monoidal infty cat, however you define them.

Given a map of nice enough monoidal quasicategories $f:C\to D$ such that $f(c)$ is an algebra of $D$ for every $c\in C$, what's the standard yoga for lifting $f$ to a map $C\to Alg(D)$? Presumably there are some conditions that need to be checked? Or maybe there's no way to really do it unless you start with more data about $f$ or something?

@JonathanBeardsley There's the Barr-Beck theorem, or maybe you had something different in mind...

@SaalHardali Hm, maybe that's the best way to think about this... thanks for pointing that out.

@JonathanBeardsley Just wondering by an algebra of D, do you just mean a monoid?

Yeah.

I mean, I'm secretly just thinking about O-algebras in general, but figured I'd just ask about associative algebras.

So I was wondering

in sort of classical universal algebra, there's a monad associated with every operad

such that Alg(O)~Alg(M_O)

is there an infty,1 version of this?

I'd have to look to find a reference but I'm like 99.99% sure the answer is yes.

would it be in Lurie or somewhere else?

I would guess Lurie, Higher Algebra. Probably around the stuff where he addresses the Barr-Beck theorem, but I'm also probably a very bad person to ask.

mhm!

and other question was when you use operad in the more expansive sense of multicategory, algebras are more complicated things, right?

Well, I dunno, an algebra over an operad is usually quite a bit of data either way. But if you've got a, say, symmetric monoidal category C with underlying symmetric multicategory MC, and another symmetric multicategory N, then an N-algebra in C is just a functor of symmetric multicategories N-->MC

mhm! thanks!

But maybe I really should just say yes.

I mean, if you've got an operad O in May's sense, then you have an underlying symmetric multicategory with one object, and so the map O-->MC picks out a single object of C with O-structure.

Yep, I was aware of it for the single-object case

it looks just like how you can represent a G-action as a map BG->C

OTOH there are symmetric multicategories with more than one objects, like, e.g., the one describing the structure of a commutative monoid and a module over it, call it Mod, say. Then a map Mod-->MC picks out two objects, one of which is a commutative monoid, and the other has an action by that monoid.

So, yeah, I suppose things can get quite complicated in that an arbitrary multicategory can pick out a bunch of objects with some prescribed relationships between them.

Very cool stuff on the arXiv digest today by the way. The Moeller-Vasilakopoulou paper, and the Klang-Kupers-Miller paper.

I wonder if anyone has tried to show that Rezk's Theta spaces are equivalent to infty,1 algebras for the terminal globular operad

I don't know how easily you could generalize the globular operad machinery to infty,1 operads

Can you get a usual, simplicially enriched operad from a globular operad? I don't know anything about gobular operads.

nah, globular operads are weird. There's a free strict omega-category monad T on globular sets, and you look at the category of globular sets sliced over T(1), which can be made a monoidal category

called Coll

and globular operads are monoids in this monoidal category

Ah.

Tom Leinster's book explains what's going on here

Yeah I figured.

in an ordinary operad, you have operations indexed by the natural numbers

but in the globular operad setting, you think of the indexes as the fibres over each pasting shape, which are all elements of T(1)

I see.

Right. I remember reading something about this a while ago. Are opetopes involved?

but the reason why I ask is that none of that seems to be un-homotopical

nope

you know the shapes that appear in Theta

Ah ok.

those are the indexing shapes

I see.

to correct myself, I mean to say that none of the constructions you're doing seem to be "un-homotopizable"

replace globular sets w globular spaces

I think the free strict omega-cat monad basically extends to 'free strict simplicial object in omega-cat'

and you can do the slicing and all of the rest is coming from universal constructions

so at least the definition of a globular operad should just make sense directly

and then algebras for the globular operad can be defined to be algebras for the associated monad, but instead of doing it in globular sets it's going to globular spaces, and that's an infty,1 cat

but yeah, I don't know if anyone's looked at this

I see. Yeah I definitely haven't heard anything about it.

Is the globular category a generalized Reedy category?

It's an actual reedy category, isn't it?

I have no idea.

But it sounds like the answer is yes. So you get good model structures on things.

yeah, so the category Theta is the syntactic category for that terminal globular operad, so I wonder if being a homotopy-algebra for the terminal globular operad can be shown to be truly equivalent to a cellular space

maybe I'll look into it

@SaalHardali We were talking about that localization of Fun(Tree^op,T) for a topos T, and I was thinking about it for a bit. If you just localize at the Segal core inclusions, then yes, I think you do get internal operads, but I was thinking about what the 'complete' part is saying, and I think you do actually get the enriched notion.

@HarryGindi I'm skeptical about that. For Segal objects this is not true (you also have to add the condition that the object of 0-simplices is a constant sheaf)

@DenisNardin so what are complete Segal spaces in a topos T for T not spaces?

Internal categories?

I don't think so. Internal categories should be the non-complete variant

Well, I'm very far from an expert on internal categories but let us take the examples of the topos P(O_G) where O_G is the orbit category of a finite group

Then a complete segal object there is just a functor O_G^{op}→Cat_∞

At least classically, one way to define internal categories is literally as simplicial objects satisfying the strict Segal condition

going way back

Every P(O_G)-enriched category gives such a functor, but the functors arising this way are exactly those for which the composite O_G^{op}→Cat_∞→Gpd_∞ is a constant functor (where Cat_∞→Gpd_∞ is taking the maximal subgroupoid)

so maybe the definition of the completeness condition is incorrect and needs to be made relative to T?

because what is just a plain Segal object in P(O_G)?

I don't know, but a complete Segal object is what I'd call a "category object" in T

is the completeness condition automatic in the classical case?

No

so it's weird then that the notion of internal category should be different between the classical and infty,1 cases is all

I don't really understand the classical one, but adding completeness is equivalent to (homotopically) invert fully faithful and essentially surjective morphisms

I'm very very hazy on this things, but I thought that even in the classical notion you wanted to invert FFES morphisms. Maybe I'm wrong

it's an interesting question. Maybe if I ping @mikeshulman he might mysteriously appear?

I'll look into it too, but maybe you're right

@DavidRoberts might also know

Actually let me retract the statement above: I don't know of a characterization of the functors P(O_G)→Cat_∞ arising from an enriched category, but they're still not all of them

According to the nLab, Lurie defined internal categories to just be Segal objects, but the nLab article makes the argument that you make that in fact we want to look at complete Segal objects

More or less, I want internal categories in a sheaf topos to be sheaves of categories. But YMMV :)

Yes, the nLab calls Segal objects internal precategories

so I guess in order to define enriched things, you need to know what 'constant' objects of T are, which makes sense when T is a Grothendieck topos

Well, you can define enriched things in much more generality than topoi

yes, that was the question I was leading to

what is a 'constant' object in a symmetric monoidal infty-cat?

I guess it's not so clear, and this is why Rune's upcoming paper is more delicate and less obvious than in the case for T a topos

I mean, internal categories are in a sense a more restrictive definition than enriched categories because I don't think they make sense unless the symmetric monoidal structure is cartesian

While enriched categories work with pretty much any symmetric monoidal structure

True. I wonder if Rune knows whether his version of enrichment agrees with this notion of T-enrichment for T a topos

I think you can reduce quite easily to the case of a presheaf topos, but then I'm a bit fuzzy about what should be true

I'd look into it but I don't think Rune has put up a preprint yet

I must have missed something. A preprint about what?

@RuneHaugseng There is a naive way to define a T-enriched category for a topos T using complete Segal objects whose X_0 is constant. We were wondering if your notion of enrichment agrees with this naive one when the symmetric monoidal infty,1 category in question is a topos

@DenisNardin He gave a talk at Cambridge about a new way of defining enriched infty categories

If he has an E_n-monoidal infty,1 category V, then he can define an E_{n-1} monoidal infty cat called Cat_V, if I remember correctly

I may have gotten the direction wrong, it might be enriching in an E_{n-1} monoidal guy makes Cat_V E_n monoidal. I don't remember exactly.

newton.ac.uk/seminar/20180705133014301

there is a video of the talk there

@RuneHaugseng By the way, I just rewatched this talk, and it looks like the thing I was talking about was the infty-cat V_\otimes. When V is an ordinary small monoidal category, this is equivalent to Theta[V].

Really Theta[-] is defined for any promonoidal structure on a category V. In the case where the promonoidal structure is representable (i.e. V monoidal), it has a simpler description

Rezk's Theta[-] takes a category C and the promonoidal structure that always exists for C (the cartesian product of C-presheaves) and uses that implicitly

So I was saying that it looks a lot like the V_\otimes construction is the derived functor version of Theta

and in the case where the infty-cat V admits a monoidal model presentation as the localization of simplicial presheaves on some small category C, you should be able to give a model category presentation of Cat_V using Rezk's machinery

In particular, working a bit harder, you should be able to show that your construction of Cat_{infty,n} is equivalent to Rezk's Theta_n model

So I expect in the most general case where V is a locally presentable monoidal-biclosed infinity,1 category, which means it admits a presentation as the left Bousfield localization of simplicial presheaves on C such that sPsh(C) admits a homotopy-monoidal biclosed structure in the sense of Heuts, Hinich, and Moerdijk

Then the 'generalized Rezk model structure' on sPsh(Theta[C]) should model Cat_{∞,1}^V

and if the homotopy-monoidal biclosed structure is symmetric, you can get the Cat_{infty,n}^V for all n by iterating

Although going too general might make the proofs a lot harder (probably working with presentations where the homotopy-monoidal structure on sPsh(C) is not representable by an honest tensor product would be where things become dramatically more difficult)

At any rate, it might not be up your alley, but it would prove that your construction is the fully infinity,1-ized version of Rezk's construction

At any rate, to prove that it is a strict generalization, you only need to treat the case where the product is symmetric monoidal closed on the nose

@HarryGindi It sounds like you're talking about the description of an E_n-monoidal infinity-category as a fibration over Theta_n^op? The construction I described used Delta^n, but you can definitely do the same thing with Theta_n instead, and it should be equivalent.

Not exactly, I was only talking about the n=1 case, but there is an iteration that goes on

Theta_n is just Theta[Theta_n-1]

Each time you iterate your construction, you are effectively getting another Theta[-]

At any rate, the case n=1 is the important one, since you can apply it iteratively. I'll write up a very precise conjecture that would prove that your construction strictly generalizes the Rezk construction and e-mail it to you before the end of the summer.

If indeed they are the same, it suggests to me a conjecture: if V is symmetric monoidal closed. the induced symmetric monoidal structure on Cat_{∞,n}^V should also be closed. This would generalize Rezk's proof in the case where the monoidal structure is the cartesian monoidal structure