@TylerLawson In Gepner-Groth-Nikolaus' Universality of multiplicative infinite loop spaces machines, Example 7.2.(ii), they give a definition of E_n-ring space. Would that be it? What are the tricky things about E_\infty-ring spaces you're alluding to? I just don't really know that theory.

Also, this is perhaps naive, but... since E_infty-groups are connective spectra, why aren't E_n-ring spaces (in the sense of GGN) just connective E_n-ring spectra?

@BrunoStonek Im happy I was helpful but this was probably only luck. I have no idea what a E_infty ring space is. Is it a different notion than a commutative monoid in spaces? what is it then?

one definition (the one in GGN, the reference above) is that it is an E_\infty-algebra in E_\infty-groups

with the monoidal structure which corresponds to the smash product of connective spectra, if I'm not reading it wrong

more classical definitions can be found in this paper by May arxiv.org/abs/0903.2813

FWIW, I also think that GGN definition of E_n ring is the same as connective E_n-ring spectrum.

Which seems pretty reasonable if you posit that an E_n ring has an "underlying additive operation" which makes it into a grouplike E_oo-monoid. (I'm not sure that this seems like the most natural requirement to me, but I also know very little about these things in practice.)

This is also Theorem 9.12 (last sentence) in May's paper, if I read it correctly.

that's reassuring

@TomBachmann If this is the definition then I know what it is, thanks :-) I was quite troubled with the idea that there's another notion going around (as it is the only natural thing I could come up with), and its good to know its not the case.

I would wonder if the category of E_n-spaces is E_{n-1}-monoidal under smash product, and if I can then take E_m-algebras in E_n-monoids, for m<n. Thus giving a bi-infinite family of notions of rig space.

I idly wondered about that as well. it's a similar question to what's going on in that big diagram in Barwick's operator categories paper, if I remember correctly

I mean the one in page 41 here arxiv.org/pdf/1302.5756.pdf

that one's about different degrees of associativity vs. different degrees of commutativity, though

here's a vague question. how can one do so much with simplicial commutative rings when the underlying simplicial abelian groups are just generalized Eilenberg-Mac Lane spaces? seems awfully restrictive

wrt to the original question, I'm not sure E_n-ring spaces (in the GGN sense) are really what fits in the analogy: shouldn't it rather be spaces which have a strict addition and an E_n multiplication?

@BrunoStonek also rings have underlying "just abelian group" which is very restrictive, still commutative algebra is cool. Isn't it a direct analog?

more precisely, the situation you are bothered with is obtained from the one I described by doing "animation", and I guess this construction preserve interestingness difference :-)

@S.carmeli perhaps I should have said: how can one do so much homotopically flavored stuff with SCRs when additively they're homotopically so easy

Probably a real homotopy theorist would do better job answering, but just note that there's nothing special about SCR-s here: also connective E_infty algebra over F_2 has very simple underlying space but it gives a lot of homotopical information (e.g. the Adams spectral sequence...). This is, classically, what algebraic topology is about, isn't it?

I'm new to these things, but it seems to me that when discussing stacks in algebraic geometry there is a little bit of a tension between the natural choice to consider sheaves on all affine schemes and the fact that this leads to an unpleasantly large $\infty$-category

One middle ground is to only consider presheaves $Aff^{op} \rightarrow \mathcal{S}$ of spaces which are small colimits of representable presheaves (I think these are sometimes called in the literature "basically bounded" or "small"). One can show that this leads to a complete, cocomplete $\infty$-category which is "class-presentable" (ie. is generated under colimits by a large set of small objects).

Moreover, I found an old result of Waterhouse which tells you that any such basically bounded presheaf admits a sheafication with respect to the flat topology which is again basically bounded, in particular valued in small spaces (I found this proven for sheaves of sets, but I would expect there is no difference if one works with spaces instead).

Is there any place in the literature where this approach to stacks is discussed, and basic results are proven? I know this is roughly how Clausen-Scholze define condensed sets, but that's quite a different context.

anyone know the password (or where to get it) for the MSRI seminar Zoom?

I would also very much like the password

Seems like we're out of luck Jeremy

They seem to be posted here, with a few days delay: msri.org/videos/recently_uploaded

@PiotrPstrągowski I'm not sure if this is really what you're looking for, but you might find some of the material discussed in arxiv.org/pdf/1904.09966.pdf#subsection.1.4 useful.

Tomer's talk is on youtube here: youtube.com/watch?v=_4gUNZ4e8os