@DylanWilson I guess the general version is that you can take the envelope of a generalized ?-operads to a cocartesian fibration over ? that satisfies the Segal condition for various ?, including Theta_n^op. What's special about ?=Fin_*,Delta^op is that this takes ?-operads to ?-monoidal infinity-categories.

As you say there is also a left that takes a ?-operad to a ?-monoidal infinity-category, but I don't know if this has an explicit description over Theta_n^op for n>1. I suspect this is related to the additivity theorem being non-trivial somehow...

For one thing, the condition that the categories of active maps to x satisfy the Segal condition in x is closely related to there being a simple description of free algebras. Using Theta_n^op you have such a description of free n-categories for all n (the "big" Segal condition) but not for free (E_n-)monoids for n>1 (and if you did you could prove additivity by a simple comparison of monads).

@RuneHaugseng Okay wow, so this is super weird. Should we not then say that "cocartesian fibrations over ? are ?-monoidal ∞-categories?"

Like, if we work in Barwick's setting, we can describe ?-∞-operads as certain things with nice maps to ? (where ? is the Leinster category of some operator category), but we cannot describe ?-monoidal categories in the same way as Lurie does for general ?.

@JonathanBeardsley I'm not sure what you mean by that - from any perfect operator category you get two Segal conditions (or two "algebraic patterns" in the terminology of my papers with Hongyi): one where you only look at inert maps to the point and one where you look at inert maps to all objects that receive an inert map from the point.

You can define analogues of operads and monoidal categories using both, and all four classes are useful

For Fin_* you get (symmetric) infinity-operads and symmetric monoidal infinity-categories if you just look at the point, and what Lurie calls "generalized infinity-operads" and something that maybe doesn't have a name (but corresponds to commutative monoids in slices Cat/C).

For Delta^op you get non-symmetric operads and monoidal categories if you take the "small" Segal condition with just the point [1], and generalized non-symmetric operads and double categories if take the "big" Segal condition if you take [1] and also its "subobject" [0]

In general you might consider any "algebraic pattern" which is an infinity-category O with a factorization system (called inert and active maps) and a collection of "elementary" objects; then you can define "Segal O-objects" in any infinity-category C with appropriate limits as functors F: O -> C such that F(x) is the limit of F over inert maps from x to elementary objects.

We can then define "Segal O-fibrations" as cocartesian fibrations corresponding to Segal O-objects in Cat, and "weak Segal O-fibrations" (for lack of a better name...) as the analogue of infinity-operads over O (functors to O with cocartesian morphisms over inerts, etc.). Looking at the full subcategory Env(O) on active morphisms in Fun([1], O), evaluation at 1 gives a cocartesian fibration Env(O) -> O (this just uses the factorization system)

Now you can ask whether this is a Segal O-fibration, i.e. whether you can glue a compatible family of active morphisms to the objects in the elementary decomposition of some x into an active map to x. Sometimes it is and sometimes it isn't. This is clearly necessary to have a simple "enveloping Segal fibration" functor on weak Segal O-fibrations, and I suspect it's also more or less sufficient (but I haven't really thought about this).

As I said above, on the Leinster category of a perfect operator category there are two natural algebraic pattern structures. For Delta^op and Fin_* you get that Env satisfies the Segal condition for both, but for Theta_n^op and (Delta^op)^n with n > 1 you only have the Segal condition for the "big" pattern.

@RuneHaugseng ah, hm... okay I suppose I'm not being clear. I think I'm just asking what kind of structure (if any?) should or can be used to define "braided monoidal ∞-categories" as something living over θ₂, maybe as cocartesian fibrations satisfying SOME Segal condition, akin to using cocartesian fibrations satisfying a Segal condition over Fin_* to define symmetric monoidal categories.

That's exactly what you get from the "small" Segal condition

Corresponding to functors F: Theta_2^op -> Cat that are Theta_2^op-monoids, in the sense that the value F(X) is the product of F([1](1)) over the inert maps from X to the 2-cell [1](1), i.e. over the number of 2-cells in X

These are equivalent to E_2-algebras in Cat

But the only way I know to prove that is using the additivity theorem

I see, got it.

I think I was just confused about " the envelope of Theta_2^op doesn't give you a Theta_2-monoidal structure on the active morphisms, but something a bit more general." But I think I just don't know what a Theta_2-monoidal structure is anyway. I'll look at your paper with Hongyi.

This is what I mean by a Theta_2-monidal structure

*monoidal

You don't get this on the active morphisms because the fibre at [1](0) is not a point

You have to somehow "contract the edges" in 2-dimensional pasting diagrams to go from the active morphisms to the free E_2-monoidal category

Suppose I have an accessible $\infty$-category $A$, and a natural transformation $f\colon F\to F'$ of accessible functors $F,F'\colon A\to \mathcal{S}$. Let $A'\subseteq A$ be the full subcategory of objects $a$ in $A$ such that $f(a)$ is an isomorphism. Is $A'$ also an accessible $\infty$-category? (I'm pretty sure it is, I can prove it, but I want to make sure. Also, is there an explicit reference?)

Proof: 5.4.6.6 says the category of accessible categories/functors has pullbacks (which are as computed in categories). This $A'$ is an example of such.

So, suppose I have a principal G-bundle E→B with E contractible. Then it's easy to show that G=ΩB. But what's a simple argument to show that they are equivalent as groups?

@DenisNardin is it circular to say that the bundle is classified by a map B-->BG? (and then you get a group map upon taking loops)

I guess to avoid that, you could argue that $E^{\times_B \bullet +1}$ has a shearing map to $B_{\bullet}G$ which is an equivalence and this gives the equivalence $B\simeq BG$

(er, rather, you've got more naturally a zig-zag between the simplicial objects that are like "E^{\times_B \bullet+1}", "G^{\times \bullet}\times E", and "G^{\times \bullet}")

This really depends on how you define "$\Omega B$ is a group".

@Denis I agree with Dylan and Charles that it's a little bit about terminology, but to add to what Dylan said, the simplicial object $G^{\times \bullet} \times E$ encodes the action of $G$ on $E$. This is a groupoid and since E is contractible, it is a group in the sense of HTT 7.2.2.1. Since all groupoids are effective, this is canonically equivalent to the Cech nerve of $E \rightarrow B = |G^{\times \bullet} \times E|$, but the latter is the simplicial object presenting $\Omega B$ as a group.

Well, I meant that ΩB has a canonical E_1-group structure, I just want the equivalence G=ΩB to be an equivalence of E_1-groups (and yes, it is secretly to prove that every principal bundle is classified by a map to BG)