Anyone have a compact description of the monoidal envelope of the associative ∞-operad?

@JonathanBeardsley Finite ordinals (a.k.a. $\Delta_+$). The monoidal operation is given by concatenation

he's still suffering from jet-lag @DenisNardin

btw, shouldn't "semisimplicial" be hyphenated, otherwise the abbreviation "c.s.s." doesn't make sense?

@user2646 It used to be hyphenated. I don't think anyone uses the hyphen anymore, so I just silently dropped it

yeah, Mike dropped it also

thanks for the clarification

("silently" dropping things is a pet peeve of mine :)

@DenisNardin Ah sure, of course.

@user2646 fantastically creepy.

So suppose that $A$ is a monoid in a quasicategory $C$, and $M$ is an $A$-module. I want to say that this data is contained in an "action groupoid" diagram in $C$, which should be a simplicial objects in $C$. In particular, I would like to say that given a suitable action groupoid simplicial object, then I can conclude that $M$ is an $A$-modules. Does anyone know how to do this?

@JonathanBeardsley at a first glance it seems like HA.4.2.2 might be relevant?

@TomBachmann So that gets at the general idea, but the issue there is that it only seems to work in the case that $C$ is Cartesian monoidal. In particular, unless I'm mistaken, the underlying "algebra" in that context is going to be a monoid in $C$, which is an algebra with respect to the Cartesian monoidal structure.

Where I use the term "monoid" in the sense that Lurie uses it in 4.1.2.5.

I.e. a simplicial object such that $F([n])$ is equivalent to the $n$-fold product of $F[1]$.

Yeah I guess that's right. I suppose the usual trick is to work in the cartiesian monoidal oo-category of oo-categories, then get a simplicial description of a C-module category, and then somehow go from there. Don't quite see how to do it, though.

Hmmmm...

I see. There may be something to that notion, by somehow using the multiplication (and action) maps of $C^\otimes$ to "project" to $C$ or something, if I'm understanding correctly.

@user2646 yeah, that is really creepy.

But still, this somehow goes the wrong direction in that we're getting the simplicial object from the existence of an action, which I think is kind of the "easy" direction.

I would really really like an identification of certain simplicial objects with the category of associative algebras, and another identification of certain simplicial objects with $LMod^\otimes$-algebras.

I mean, given an algebra $A$, i.e. a section of a coCartesian fibration $C^\otimes\to Ass^\otimes$, one can take the monoidal envelope of $Ass^\otimes$ to obtain a map $\Delta_+\to C$ classifying the algebra. Further, I seem to recall there being some kind of "decalage" map $\Delta^{op}\to \Delta_+$ and composition with this gives me the bar construction on $A$, i.e. the "simplicial object describing $A$." But it's not clear that this process is necessarily invertible.

It may be something like taking some kind of Kan extension along that decalage map.

Actually I'm sorry that's wrong.

The decalage map defines not the bar construction but the Amitsur complex for the algebra (I think).

Actually, haha, I don't even think I want to use decalage here. It might just be the inclusion $\Delta\hookrightarrow \Delta_+$

Yeah okay so actually I don't know anything anymore...

I suppose there might be some kind of Koszul duality statement to be made here. I mean, the simplicial object associated to an augmented monoid is really the simplicial object that picks out the coalgebra associated to that algebra by Koszul duality. So that would probably be the way to make things precise.

wait, I'm still confused why definition HA.4.2.2.2 isn't what you're asking for. It looks like it describes a module over a monoid in terms of an 'action groupoid'; i.e. the M' in that definition appears to be what you're after, yeah?

@DylanWilson as far as I can tell, this only works for a Cartesian monoidal category though, right?

In particular, the algebra must be over a monoid

Given in 4.1.2.5

And a monoid is an object $X$ with a map $X\times X\to X$

Moreover, the "action maps" described in 4.2.2.2 are all, basically, of the form $M\times X\times\cdots \times X\to M$, where those are all products.

Ah, I see. In general I suspect what you're asking for is to identify the monoidal envelope of "LM^{otimes}" with like some category related to Delta. Saul did this for modules over commutative algebras, so maybe you could mimic that argument. Probably a version of this is contained in work of Ayala-Francis, more generally.

see also 2.2.6 in DAG II

(in HA maybe that's like Variant 4.2.2.11 and below?)

@DylanWilson Yeah very possibly, though I'm inclined to think that the monoidal envelope of $LM^\otimes$ will look more like $\Delta_+$, giving a sort of cosimplicial presentation, but maybe that's the right way to do it anyway.

The more I think about it, the more I think that trying to describe an algebra simplicially is kind of dumb, and is really just a Koszul duality statement.

It's naturally a cosimplicial thing.

Ah but actually... wow...

hmm... I guess i was convinced by 4.2.2.11 that the simplicial point of view works

4.2.2.11 looks very much like what I'm thinking...

Hmmmmmmm....

nice!

Okay, haha, well in some sense this further confuses me. I had gotten myself to the point of feeling okay that I needed to think cosimplicially...

@DylanWilson actually I think it's not quite right still. Notice that the thing that $\Delta^{op}$ is mapping into is effectively $C^\otimes$ (well, it's a correspondence with $C^\otimes$ at one end). In other words, this isn't really any different than noticing, from DAG II, that we can work with either $Assoc^\otimes$ or $\Delta^{op}$ as our "associative $\infty$-operad,"and algebras are just sections of a cocartesian fibration.

The issue of course is that a map $\Delta^{op}\to C^\otimes$ doesn't give a map $\Delta^{op}\to C$, does it?

Maybe there's a ways of smushing down everything to get a honest simplicial object in $C$ though?

But even so, this seems like it'd be quite hard to prove that such a smushing operation determined an equivalence.

@JonathanBeardsley To do that you really need C to be a cartesian symmetric monoidal category

@DenisNardin Okay, yeah, that's what I thought.

That's fine though. I'm pretty happy with the "take the monoidal envelope of $Ass^\otimes$" approach.