@MaximilienPéroux I think you might be confusing yourself by just writing C^op: if C^\otimes -> O is O-monoidal then we're looking at the dual fibration C_\otimes -> O^op and its opposite (C_\otimes)^op = C^{op,\otimes} -> O; if you apply that to id_O you just get id_O again, not "O^op" (which doesn't make sense if O is just an operad and not symmetric monoidal).
So Fun^{oplax}_{/O}(O, C^\otimes) = Fun^{lax}_{/O}(O, C^{op,\otimes})^op = coAlg_{O/O}(C) as you say
Yes Exactly, O^op made no sense to me. Ok I understand that the opposite O-monoidal structure of O is O itself is what you are saying. Ok that makes much more sense, that cleared everything up I think, thank you :)
A group G is said to have FL property if BG is homotopy equivalent to a finite CW-complex. Take K a torsion free group. Is K a filtered colimit of FL groups?
let $R$ be a commutative ring, and consider the symmetric monoidal category $(Ch_R,\otimes)$. are there characterizing conditions known for when this is a presentation of the $\infty$-category $Mod_{HR}(Spectra)$? i'm guessing the answer is "iff $R$ is a $\mathbb{Q}$-algebra", based on the corresponding decategorified fact about $E_\infty$-algebras.
the latter i have always imagined is proved using an $E_\infty$-operad in $Ch_R$, which is quasi-isomorphic to $Comm$ just when the symmetric groups have trivial $R$-cohomology. assuming that's correct, i'd be interested to know a proof-sketch for the former along the same lines.
@epic_math i'm glad to hear that! i am so grateful for this chatroom. it has made all the difference in the world for me, throughout my career, to have easy access to a community of people with similar math interests.
@AaronMazel-Gee I would've expected the answer to be 'always', but maybe I'm misunderstanding something. It seems like $(Ch_R, \otimes)$ presents a stable, presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and that the endomorphism algebra of the unit is $R$, this produces a functor $\mathsf{Mod}_{HR} \to \mathcal{C}$ which is a symmetric monoidal enhancement of the usual one, which is an equivalence.
are the categories of semisimplicial sets and Delta-complexes equivalent? (a morphism of Delta-complexes must carry each n-simplex to an n-simplex by the distinguished linear homeomorphism that preserves the ordering of the vertices.)
@DylanWilson ohhhhh i think i see my mistake! thank you. once again, i'm not even sure how i confused myself on this point.
i figured that the failure of commutative algebras in $Ch_R$ to present all of those in $Mod_{HR}$ was coming from a categorified failure of the symmetric monoidal structure to be "correct".
in other words, i was misunderstanding the fact that $Alg_{E_\infty}(Ch_R) \simeq CAlg(Ch_R)$ iff $R$ is a $Q$-algebra. so maybe a better interpretation is that for model-categorical reasons, $Alg_{E_\infty}(Ch_R)$ is always a presentation of $CAlg(Mod_R(Spectra))$.
@DylanWilson great, thanks! i think the answer to my question is "yes": Delta-complexes are just a topological way of defining a category that's equivalent to semisimplicial sets (meaning functors $(\Delta_{inj})^{op} \to Set$). in both cases, one sees the failure of the definition to encapsulate "natural" maps, e.g. $hom_{\Delta Cx}(\Delta^m,\Delta^n)$ is empty for $m>n$, which leads to simplicial sets.
oh haha it turns out hatcher even has an appendix spelling this out, which is in the online version but not the print version (which is why i missed it -- trying to cut down on my screen time while preparing lecture notes).
