@CharlesRezk Do you mean a presentation of the (degree >=1) part of the operad using free cells? (I don't know of one.)

These aren't the 'permutohedra'?

Given a simplicial functor $F\colon S \to \mathbf{Cat}_{\infty}$ which comes from a collage $*+S \hookrightarrow Q$, is there a formula for the lax colimit of $F$ in terms of the collage?

@EdoardoLanari What's a collage?

There is an equivalence of categories between diagrams $A\to \mathbf{sSet}$ and the full subcategory of $\ast + A/\mathbf{sSet}-\mathbf{Cat}$ on those maps $F\colon\ast+A\to Q$ which are bijective on objects, fully faithful when restricted to $\ast$ and $A$, and have the property that there are no arrows in $Q$ from the image of $F$ to $\ast$.

Essentially, I am trying to make sense of Lurie's straightening functor in this language, since given a map $p\colon X \to S$ its straightening $\mathfrak{C}(S)\to \mathbf{sSet}$ comes from a collage $\ast + \mathfrak{C}(S) \to \mathfrak{C}(p)$. Thus, morally, $\mathfrak{C}(X)$ should be its lax colimit and I wonder if this can be seen already in $\mathfrak{C}(p)$ (the pushout that defines the straightening), so as to justify the definition.

Is $+$ the join? Or the disjoint union? Sorry, I am really unfamiliar with the notation you're using

Disjoint union. This sort of stuff is explained in Riehls and Verity's "The comprehension construction", where they adopt a different point of view on the (un)straightening business, and I have always wondered if a connections as above can be stated so as to better justify Lurie's definition (which,imho, is nice since it adapts to higher dimensional contexts, i.e. locally cartesian fibrations of scaled, but lacks some motivation).

@TylerLawson Yes, basically. Really, I just want to understand the inclusion $O_{n-1}(n)\to O_n(n)$, where $O_{n-1}\subset O$ is the suboperad of a cofibrant (non-unital) $E_\infty$-operad generated by the stuff in dimensions $<n$, as a map of $\Sigma_n$-spaces.

$\partial K_2:=O_1(2)$ is empty, so $K_2:=O_2(2)$ is the cone on this which (up to equivariant weak equivalence) is a point fixed by $\Sigma_2$. $\partial K_3=$ three points with transitive $\Sigma_3$-action, and $K_3$ is the cone on this.

$\partial K_4=$ the 1-skeleton of a 3-simplex, together with three additional edges which connect midpoints of opposite edges (but which don't meet in the middle). The $\Sigma_4$-action is the one permuting vertices of the tetrahedron.

In all the above cases, $\partial K_n\approx |P_n|$, where $P_n$ is the poset of (non-trivial, non-empty) partitions of set of size $n$, though the triangulation of $P_n$ does not match up well with any obvious cell decomposition of $\partial K_n$. The conjecture is that this is true for all $n$.

All above is up to equivariant weak equivalence. What I'm actually describing is the filtration for a cofibrant model for the commutative operad, where "cofibrant" means "built by attaching free operad cells on symmetric sequences which are not required to have free symmetric group actions". Take product with an $E_\infty$-operad to get a filtration for that.

@CharlesRezk morally Bar(E_{infty}) is supposed to tell you about a 'minimal operadic cell structure' on E_infty, and we know the partition poset complexes appear there. Is it possible to turn this "moral" argument into a real argument?

@DylanWilson. You mean Michael Ching's bar construction? Yes, it seems you ought to be able to do that. Though TBH I don't really understand his proof of the relation to the partition complex.

@CharlesRezk I just mean the the (derived) relative tensor product 1\circ_{E^{nonunital}_{infty}}1 (which is presumably what Ching's thing models?). as for the relation to partition posets, maybe Remark 12 here is helpful? math.harvard.edu/~lurie/ThursdayFall2017/Lecture15-Koszul.pdf

The derived relative tensor product definitely gives you partition complexes, when you build it as a two-sided bar construction. Ching's bar complex is a different construction, which is designed to give the thing the structure of an operad. Ching says they are equivalent up to homotopy, though I find the proof in his original paper a little lacking on details (maybe there's another proof elsewhere I dunno).

Anyway, the combinatorics of Ching's bar construction is similar to combinatorics you can use to identify Stasheff cells for A-infinity (I think Boardman did it that way), so its reasonable that those ideas give what I want.

Morally (and maybe explicitly), the spaces in Ching's bar construction are supposed to be the quotients $K_n/\partial K_n$ I was describing, where $\partial K_n\subset K_n$ comes from the arity filtration of the operad.

*co-operad