@JonathanBeardsley @PiotrPstrągowski Isn't this claim a consequence of the fact that CMon(S)(x)S_{\le n} = S_{\le n} (x)CMon(S) with respect to the Lurie tensor product? I mean, tensoring with S_{\le n} gives the n-truncated objects and CMon(x)C=CMon(C), so computing one way we get n-tuncated monoids in spaces and the other way we get commutative monoids in n-truncated spaces. I hope im not making a stupid mistake here.

by "this claim" i mean the claim about commutative monoids in spaces.

A somewhat more general argument uses the fact that the functor S -> S_{\le n} is symmetric monoidal.

@TomBachmann can you explain what argument you refer to?

Does anyone know a reasonably explicit geometric description of the Bott map BO×Z→Ω^8(BO×Z) for real K-theory? Ideally this would be something that eats a vector bundle of rank n on X and spits out a (difference of?) vector bundle on S^8×X...

@DenisNardin the map is given by multiplication by the class in degree 8 so it can be written as pulling pack the bundle along the projection to X and then multiplying by this specific class. This class can be written explicitely using Atiyah Bott Schapirro construction as a Z_2 graded Clifford representation, in this case the 16-dimensional rep of Cl_8 = M_R(16), so you can write a formula for it. is this what you looking for?

Hrmm... sort of. I was expecting something more geometric (i.e. built out of some manifolds etc.), but maybe I'll have to find some other way. The context is that I'm trying to write down a geometric definition of the motivic Bott class for hermitian K-theory and I'm looking for inspiration at the topological case

I might be able to cook up a motivic ABS, but that's a fairly long tangent...

It would be really fun to have a motivic ABS. There is such e.g. for equivariant real K-theory, so I hoped that motivically it exist as well, but im probably naive :-)

Oh there probably is. I even have ideas on how to approach it, but that's for a different paper hopefully :)

@S.carmeli I think that if C is presentably symmetric monoidal then C -> C_{\le n} is symmetric monoidal. This implies that the left adjoint of Alg(C)_{\le n} -> Alg(C), when viewing Alg(C) as a category of partially cartesian sections, is computed objectwise; in particular commutes with the forgetful functor. Here Alg(...) can refer to any oo-operad, I guess.

@TomBachmann I am confused about your argument. What exactly is it that you prove? that an algebra in C_{\le n} is the same as an algebra in C with underlying in C_{\le n} or that this is the same as n-truncated objects in algebras?

@JonathanBeardsley I think what you want can be shown as follows in the presentable case: we have LMod_A(tau_nX) = LMod_A(X(x)S_n). But X(x)S_n is an X-algebra in Prl and so by the identity LMod_A(M)=LMod_A(X)(x)M for every C-module M we have:

LMod_A(X(x)S_n) = LMod_A(X)(x)_X (X(x) S_n) = tau_nLMod_A(X).

sorry for the horrific notation :-D

I prove that truncation of algebras is the same as truncation of the underlying spaces

Ok now I got it, thanks!

@JonathanBeardsley I think might be a technical detail? What Piotr said seems to be getting at the equivalence being with pointed connected spaces, and it's easier to be truncated in pointed connected spaces

(than in regular spaces)

let p be an odd prime. what are the (homotopy-coherent) $C_p$-actions on the sphere spectrum, is this known?

Oh I actually didn't know that truncated objects of a category (presumably presentable?) could be obtained by tensoring with truncated spaces. That's super useful.

I didn't know this either. I think one can prove it by noting that Fun(C, D)_{\le n} = Fun(C, D_{\le n}) and then using the usual trick of writing the lurie tensor product as limit preserving functors.

What is this saying? $S^1\times X$ is not usually a 1-truncated space.

In the category of presentable categories, (n-truncated spaces) tensor C = (n-truncated objects in C)

Ah ok.

asking for C to be a module category over n-truncated spaces means that the functor C |-> X ⊗ C factors through the "unit" functor Spaces -> n-truncated spaces. so e.g. Map(S^{n+1}, Map(X,Y)) ~= Map (S^{n+1} ⊗ X, Y) ~= Map(X,Y) which implies that the mapping space is n-truncated.

I get that. Didn't follow the previous conversation.

ah. sorry

s

"n-truncated" is a kind of algebraic structure on an object.

@CharlesRezk yes. Its an algebraic structure which is at the same time a property. We call such structures "Modes" and, following @AaronMazel-Gee suggestion, ill, shamelessly, point out that we work out several examples and general structural results in the Ambidexterity and Height paper. Examples of such "modalic" properties are higher semi-additivity, chromatic height, stability, e.t.c. Each classified by a mode (higher commutative monoids, E_n local spectra, Spectra resp.)

I meant Ln^f local spectra

In the special case of S_n, its a special case of the fact that if M is a mode and M' is a symmetric monoidal localization of M then M' is a mode which classifies the property of having M'-local M-enriched hom spaces. Tensoring an M-module with M' then pick the objects with M'-local M-enriched Yoneda embedding

but of course the special case is quote straight forward and don't require all this theory, I just feel like this is a useful point of view on properties of presentable categories in general so I said it out :-)

quite

@S.carmeli This feels super helpful for my intuition!

@TomBachmann I learned this stuff from Tomer and since then im addicted :-)

Somehow I missed the modes paper coming out but I guess I have my evening reading settled now :)

:-D :-D

@TomBachmann the truncation map is symmetric monoidal even if the category is not Cartesian monoidal?

(I'm working with a case in which the category IS Cartesian monoidal, so it's immaterial to me, I was just surprised to see that)

@S.carmeli Do you think there's any issue with your argument for non-commutative monoids? I was actually only thinking about monoids, not commutative monoids.

@JonathanBeardsley I haven't thought about it in detail, but I would suspect so. (n-1)-truncation is the universal way of killing off maps from things like Sigma^n X, and because the tensor product preserves colimits in each variable these n-connected objects are stable under tensor products. So in forming (A_{\le n-1} \otimes B_{\le n-1}) you have killed off some (but maybe not all) n-connected objects mapping to A \otimes B, which should yield the desired formula ...

... (A_{\le n} \otimes B_{\le n})_{\le n} = (A \otimes B)_{\le n}.

But again I would have to think about if more assumptions are needed to turn this into an actual argument.

@JonathanBeardsley the first is a consequence of the fact that it's tensoring with an algebra, at least presentable. For your second question sure it's the tensor of E1 monoid and n truncated spaces and tensoring with the first gives E1 monoids and with the second n truncated objects.

@TomBachmann Ah sure, fair enough.

@S.carmeli Ah okay. Yeah I think that thinking about truncation in this way (so long as everything's presentable) is going to save me some headaches.

@JonathanBeardsley yah and I think it works in greater generality then presentable usually, but that gets into set theoretic issues I'm kinda scared of.

Yeah. I try to avoid those things.

Do pullbacks descend along effective epimorphisms in (oo-)toposes? That is, if I have a cube diagram given by two commuting squares lying over each other, with arrows pointing down between them which are effective epimorphisms, and I know that all faces except the bottom one are pullbacks, can I conclude that this last face is a pullback too?

@AdrianClough pullback along the map of the final vertices of the squares is a functor which is conservative and commutes with products so it reflects products also. This gives you what you want if you note that the pullback of square is the product in the over-topos.

over the final vertex

@S.carmeli That was a lot more elegant than what I expected! Thanks!

@AdrianClough you'r welcome :-)

Hi, I asked a question on MO which eventually led to me being recommended to try and ping @RuneHaugseng in this chat (though my question isn't specifically directed to him, and frankly I'm not sure if this question belongs in this specific chat room...) Briefly, I was wondering if there has been any (recent?) work towards making sense of / constructing the $(n+1,r+1)$-category of categories enriched in a monoidal $(n,r)$-category (where $n\leq\infty$).

I know that Gepner and Haugseng made sense of an $(\infty,1)$-category of categories enriched in a monoidal $(\infty,1)$-category which is related, but as far as I know this wouldn't encompass (for instance) enrichment in a monoidal bicategory of something of the sort

[I'm not familiar with chat etiquette so forgive me if I'm barging in rudely]