For some reason I seem to remember something like "for an E_n-ring spectrum R, the multiplication map is a map of E_{n-1}-R-algebras" is that true?

Ah yeah, I guess this is just the counit of the relevant adjunction (lifted to algebras)

Really starting to get frustrated that I have to bug people in here for this, but does someone know where to find, in Higher Algebra, the statement that an O-monoidal functor of quasicategories A-->B lifts to categories of O-algebras?

I guess this is what Kerodon will help with, haha.

@JonathanBeardsley I'm not sure that helps, but doesn't this follows from the fact that the category of ∞-operads is enriched in qCats?

That is, the ∞-category of O-algebras is exactly the sset of sections sending inert arrows to cocartesian arrows of the cocartesian fibration C^O→O representing the O-monoidal category

and a O-monoidal functor is exactly a map of those cocartesian fibrations

@DenisNardin actually yeah this might follow like... really immediately from definitions...

I'm not sure if it's exactly spelled out in HA, if you need a reference for citation purposes

Although I keep getting mixed up between $Alg_{/O}(C)$ and $Alg_O(C)$

what's the difference between the two?

i.e. maps $O\to C$ over $O$ and over $Fin$

Ah, right

er, Fin_*

Not the most unambiguous notation

And so actually I suppose I need to decide what I mean by "E_n-algebra of C"

Well, the difference to me seems to lay in what is C

I.e., is C a symmetric monoidal ∞-category or only an E_n-monoidal ∞-category?

Yeah fair enough. And hopefully they end up coinciding when you take the "underlying E_n-monoidal category" of a symmetric monoidal one.

I think this all comes out from the definitions. I just need to untangle them.

Yeah, the "underlying E_n-monoidal category" of a symmetric monoidal category $[C^⊗→Fin_\ast]$ is just $[C^⊗×_{Fin_\ast} E_n^⊗→E_n^⊗]$

So everything works out

Right. Okay... haha. Good.

I mean, not a surprise of course..

Okay I sort of get what Lurie's doing here. Given $C$ and $D$ both $O$-monoidal, we have categories $Alg_{/O}(C)$ and $Alg_{/O}(D)$ which are the $O$-algebras of $C$ and $D$. They're sections of the defining cocartesian fibrations of $C$ and $D$ that preserve inert morphisms.

Now, an $O$-monoidal functor $C\to D$ is a few things: first of all, it's a map of $\infty$-operads, i.e. it preserves inert morphisms.

Note that ANY map of $\infty$-operads $C^\otimes\to D^\otimes$ is "lax monoidal" because it takes algebras to algebras, by the obvious precomposition with the section $O^\otimes\to C^\otimes$.

Er, well any map of $\infty$-operads that commutes with the maps down to $O^\otimes$, that is.

But, if we want to say that the map $f$ is $O$-monoidal, as it's used in HA, then we need to also ask that it preserve cocartesian morphisms with respect to the cocartesian fibrations down to $O^\otimes$.

Which, IMHO, is the "right" kind of "map of cocartesian fibrations."

In particular, it induces a natural transformation of functors after applying the inverse Grothendieck construction.

If you want, a monoidal map is a lax monoidal map such that certain arrows are equivalences. For the case O=Ass, the condition is just that the canonical map FX⊗FY→F(X⊗Y) is an equivalence

And I think this "preserving cocartesian morphisms" ends up being what makes $f$ a "strong $O$-monoidal functor," though I'll have to think a bit about how that works.

In that case, this is the same as saying that the arrow {FX,FY}→F(X⊗Y) over the fold map 2_+→1_+ is cocartesian, and this is the image under F of the cocartesian arrow {X,Y}→X⊗Y (cocartesian since X⊗Y is defined as the pushforward of {X,Y} over the fold map)

Ah I see, that's really nice. Right, so it's saying that $F(X\otimes Y)$ is "the thing that uniquely plays the role of the tensor product of $F(X)$ and $F(Y)$".

Moreover... I think that the induced functor of algebra categories is also $O$-monoidal... though this seems to take some untangling of 3.2.4.4 of HA.

Ah jeez... I take that back. If the category $C$ is not itself symmetric monoidal then I'm not even sure that the $O$-algebras are $O$-monoidal.

E.g. if $C$ is $E_{j+k}$-monoidal then the category of $E_j$-algebras is only generally $E_k$-monoidal.

Is there a definition of Operad which is internal to a topos. Resembling how one can define complete segal space objects in any topos and if you do so i groupoids you get internal infinity categories. Can one do the same with Operads? If not why not? Is it a matter of technical difficulty or is there an inherent flaw in this idea?

@SaalHardali I know how to define Segal objects in any infty cat with pullbacks, but does being in a topos make it possible to define complete Segal objects?

oh, I guess yeah you can do it when you're in a topos

I'm not sure that i'm using all the properties of the Topos but the usual definition does seem to work in this context.

you need some kind of topos-like structure to define the analogue of the simplicial presheaf J x *

but yeah

btw Topos=infinity topos everywhere if it wasnt clear and same with operads

So you could try to do something like this using dendroidal sets

but instead look at functors Tree^op -> T where T is your topos

It's not as nice as the CSS = Quasicat equivalence because the model strucutre on dendroidal sets isn't regular in the sense of Cisinski, so its simplicial completion (i.e. the model in infty groupoids) is not a LBL of the injective model structure

LBL stands for ? bousfield localization?

yeah

Whats the ? then?

Left

Left?

ah great.

I'm talking about a weird technical condition

this regularity condition of Cisinski

but IF you can get something that looks like the CSS version of dendroidal sets

I'm not familiar with it. In any case, re you suggesting that one can port the definition with dendroidal sets to get a category of internal operads with all the usual structure it comes with?

then it should all work out

yeah

Aha okay i think the main issue in any case is phrasing dendroidal sets in this sort of "invariant"ish way

the proof that dendroidal sets and Lurie's infty operads are the same is kind of annoying

Saal, I don't think so

They had to make everything invariant-ish because the on-the-nose Boardman-Vogt tensor product of dendroidal sets is not associative or symmetric

so they had to redo a bunch of papers to show that it is homotopy associative and symmetric

and those proofs amount to essentially proving the prerequisites for Day's theorem

in the infty case

so Fun(Tree^op, T) should be symmetric monoidal wrt this Day product

Then you need to localize it somehow don't you?

and then I think you can just rephrase all of the dendroidal lifting conditions in terms of expressions containing the right adjoint to the BV tensor product and iterated pullbacks just like you do with segal spaces

I wish this was written somewhere nicely

Moerdijk is your guy, I think

If this really is valid it seems like a pretty nicw definition to me.

If it is possible, he could probably give you the long and short of it

I'm inclined to believe that it will work, but I'm not 100% positive

The only other way I could think of doing it is using that stuff Rune was talking about in Cambridge

Then you could actually just define a T-operad just like an ordinary operad using Gamma but instead of being an infty,1-cat, it's a T-enriched cat

@SaalHardali According to the nLab page, they have an equivalent model of Dendroidal spaces, so I think it should rock and roll

@SaalHardali arxiv.org/abs/1010.4956

Actually, I see a potential problem. If T is an infty Topos, is there any analogue for a map t->t' in the topos to be a Kan fibration

I dunno, I'd have to work it out exactly

sorry to bother you!

@HarryGindi Thanks! This article is right on the mark it seems.

@SaalHardali There are a few weird things in how they define the generalized Reedy model structure

I think there are superscripts that should be subscripts

but in the case where everything is infty-ized

I think you just do Fun(Tree^op,T)[W^-1] where W is the set of Segal core inclusions viewed as the pushforward of the segal core inclusions viewed as discrete simplicial presheaves

and then localize again for completeness in the same way as usual at the inclusion *->J also viewed as the pushforward of the discrete simplicial presheaf

whether or not the homotopy-monoidal structure also pushes forward to T-operads is unclear to me

anyway, if you end up finding a use for this generalization, let me know!

also, for anything dendroidal, you probably want to know how they relate to Lurie's infty-operads: arxiv.org/pdf/1305.3658.pdf

The theorem of Heuts, Hinich, and Moerdijk only proves the equivalence between dendroidal sets/spaces and infty-operads for the subcategories of operads without nullary operations, so not everything in sight is the way one might wish, as far as I know

@HarryGindi I think the model category stuff is a bit of a red herring here - you can define (complete) dendroidal Segal spaces just as a localization of presheaves on the dendroidal category in the infinity-category of spaces. That should go through the same way for presheaves in any infinity-topos, though there are a couple of things to check to ensure that completeness is still the localization at FFES maps.

@RuneHaugseng w.r.t. what are we localizing? In the case of segal space I think we are localizing w.r.t. the skeletal inclusions (and the walking isomorphism for completeness). Is that true? Should we expect this localization to be reflective in either of the cases?

maybe the reflective thing is related to the FFES condition you mention which I don't know what it stands for...

@SaalHardali I said it earlier when I was summing up. it's "segal core inclusions" and the "completeness map". I stated it in terms of an infty,1 localization

being precise about how to define the segal core inclusions in general because I don't know how to define a T-enriched Yoneda embedding

@HarryGindi Aha sorry I missed that. So at least conjecturally at this point of the disccusion this is a reflective localization and the local subcategory is the category of $\infty$-operads when T is the topos of infinity groupoids?

Precisely!