Does the functor sending a ring spectrum to its category of modules preserve limits?

@TomBachmann not in general. Take for example the trivial action of Cp on Q

The limit of the modules is representations of Cp and the modules over the limit is Q modules

@S.carmeli That's what I suspected, thanks! Are there reasonable classes of limits that are preserved? What about pullbacks, for example.

@TomBachmann isnt the diagram expressing P^1 as a gluing of two A^1-s gives a counter example? the gluing of the categories are sheaves on P^1 which is not modules, but maybe I'm not derived enough...

Well sheaves on P^1 has a generator, so it is modules over something, but I agree that it seems unclear why that something should be the obvious pullback, or even commutative.

So I guess I will just remember that the functor preserves very few limits other than finite products. Thanks again! :)

Right, even the derived global sections of the structure sheaf on P^1 is just k in degree 0, so that is a counterexample

In the positive direction you can find some criteria in George Tamme's article on excision: arxiv.org/abs/1703.03331

Georg, sorry

to make a generator of QCoh(P^1) you need two "adjacent" serre twists, e.g. O and O(-1)

It does also satisfy flat descent, right? So some cosimplicial limits are preserved too.

Yes sure a better way to phrase what I said is that the pullback of the rings is the global sections of the structure sheaf and the comparison map is exactly the formation of global sections so any no affine spectral scheme gives a counter example I guess.

@RuneHaugseng, @DustinClausen: Very good points, thanks!

In particular SAG.16.2.0.2 shows that certain pullbacks are preserved when only considering connective modules.

Extremely ill defined question: One can think of the category of commutative rings via their modules as a subcategory of the category of $\mathbb{Z}$-linear symmetric monoidal categories. Can something similar be done for, say, Huber pairs (and/or analytic rings in the sense of Clausen/Scholze) in the condensed setting?

Or is this somehow a "wrong" way of thinking of adic/analytic spaces?

Not quite, Schachar: actually quasi-affine schemes are also derived affine. So e.g. the diagram gluing A^2 - (0,0) from A^2 - the coordinate axes is an example not a counterexample!

P.S. i'm sorry that for some reason I keep misspelling your name!

Hi Saal, you need more structure: it should be tensored over condensed sets as well. But then it's ok.

i.e. it's not enough to know the sym mon. category, you need to know what the free objects 1[S] are on profinite sets S

this promotes End(1) to a condensed ring, and then you're off to the races

@DustinClausen I guess after few years in Bonn that's the default way to spell my name, and actually, the translation I chose was pretty random, so I don't think its actually a miss-spelling (many do shahar for example). Yes I agree, so Im probably confused about terminology, in this case A^2-0 is not considered "affine derived scheme"? I should probably stop being lazy and read SAG once and for all.

@TomBachmann isnt the statement for connective modules implies the one for non-connective? I mean this is tensoring with Sp over Sp^cn and the tensor product of Prl... likes limits or so?

In the usual terminology affine derived schemes are supposed to have connective structure sheaf. This is nice because then you can read the underlying topological space off the \pi_0 and so on.

In this example \pi_0 = k[x,y], so you seem to still see the point you deleted...

Well there is an explicit warning with a counterexample in the non-connective case in SAG.16.2.0.3...

@TomBachmann interesting ill take a look thanks!

@S.carmeli In fact if I'm not mistaken spectral schemes with the property you're thinking of are what Lurie calls "quasi-affine" derived schemes, although it gets muddier when you go to derived stacks

@DenisNardin cool thanks!

has anybody seen an infty-categorical adaptation of the following 1-categorical fact? Let C be a monoidal category with a free-forgetful adjunction (F,U), where U:C\to Set is faithful, and moreover F(*) =1, the monoidal unit of C. Then End_C(1) \cong Nat(id_C,id_C)

the proof I know of this statement doesn't immediately generalize

@DustinClausen That's what I was hoping would be the case. So to make things clear, for formal categorical reasons (and lets use Pyknotic convention for simplicity so we can use presentable categories) there's a fully faithful embedding of commutative algebras in $Cond_{\mathbb{Z}}$ into symmetric monoidal $Cond_{\mathbb{Z}}$-linear categories categories.

Then presumably general analytic rings are not too far from the essential image of this embdding. In that if you forget the $Cond_{\mathbb{Z}}$-module structure they are symmetric monoidal equivalent to modules over some condensed ring.

I feel like I need some extra condition to isolate exactly the analytic rings though...

Maybe I want that the canonical map to condensed modules over the endomorphisms of the unit is fully faithful?

not sure if that would be enough...