Okay I think I'm starting to understand conceptually why we cannot, for instance, talk about commutative monoids in a braided monoidal ∞-category. I think I'm getting at this by comparing to the case of trying to think about commutative monoids in a monoidal 1-category. But I'd still be interested to see something really precise about why, just flat out, one cannot even ask for that kind of data.

In the 1-categorical case, if one tries to ask for commutative monoids in a monoidal category, it seems that one is forced to ask for there to be a twist map (with respect to which the multiplication is invariant), and I just don't think there's any, like, canonical or generic way to pick such a map.

Whenever one talks about commutative monoids, one is secretly talking about "commutative monoids with respect to this class of twist maps."

@JonathanBeardsley I don't know of a name in the non-infinity setting (though something like this might well appear in Batanin's work on n-operads in the n=2) case. The idea is the same as for the symmetric monoidal envelope of an operad: you take Env(Theta_2^op) to be the full subcategory of arrows in Theta_2^op with active maps as objects, and map this to Theta_2^op by taking a morphism to its target. This is a cocartesian fibration, using the inert-active factorization system on Theta_2^op.

(Specifically, if f: I -> J is active and g : J -> K is any morphism, then gf factors uniquely as an inert map i : I -> X and an active map a : X -> K, and the resulting square is the cocartesian morphism over g with source f - or a = g_!(f).) The fibre Env(Theta_2^op)_I is the slice (Theta_2^op)^{act}_{/I} of active morphisms to I and triangles between them, where the factorization system forces the third map to be active too.

Now you can check this satisfies the Segal condition for Theta_2^op - this basically says if I have a compatible family of active maps to the 0-, 1-, and 2-cells of an object J, then I can glue them to a unique active map to J.

But it is not a Theta_2^op-monoidal infinity-category: the fibre at [0]() is a point (only [0]() admits an active map to [0]()), but the fibre at the 1-cell [1](0) is not a point, but Delta^op: the objects that have active maps to [1](0) are precisely the 1-dimensional ones, [n](0,...,0), which give a copy of Delta^op.

And the fibre over the 2-cell [1](1) is the active subcategory of Theta_2^op (since every object has a unique active map to [1](1)). This accounts for the fact that if I have two objects [n](a_1,..,a_n) and [m](b_1,...,b_m) then I can always glue them horizontally to [n+m](a_1,...,a_n,b_1,...,b_m), but I can only glue them vertically if n=m (so their images in Env(Theta_2^op)_{[1](0)} agree), where I get [n](a_1+b+1,...,a_n+b_n).

@RuneHaugseng weird... does something similar happen if you try to do monoidal envelopes in the "Delta^{op}-operad" setting? certainly there is some left adjoint to the inclusion of X-monoidal categories inside X-operads (where X is the Leinster category of some perfect operator category); I wonder what the description is if it's not 'take active morphisms' and I wonder why that description collapses to 'take active morphisms' in the symmetric case...

@RuneHaugseng @DylanWilson yeah that's super interesting. I think I need to go look at the symmetric monoidal situation in HA

@AaronMazel-Gee I am also teaching an "undergrad" topics class in derived categories in algebraic geometry next semester. It might be of a more "applied"/geometric bent but our standard text is Huybrechts' Fourier-Mukai book. The first two chapters are fuss-free introductions to the topics you mentioend

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