Geoffroy Horel

@PaulVanKoughnett The category of profinite sets has all limits and colimits (this is the case for the pro-category of any category with finite limits). Then simplicial profinite sets also have all limits and colimits as those are computed objectwise.

@DenisNardin I think by an EM Moore type argument you can identify this THH with cochains on the free loop space of K(Z/p,n) this free loop space splits as K(Z/p,n-1) wedge K(Z/p,n)

It seems that it's closer to being G(n) (the groupe of homotopy automorphisms of the n-sphere

@SaalHardali This is an open question. There is a rational computation due to Fresse and Willwacher that suggests that it's not O(n).

Ï don't know but I would bet against it. I think at the time I wrote this I had a counter-example (or at least a reasonable candidate for a counter-example) but I never actually wrote down the details anywhere. Now I have no idea what this conter-example was :)

@TimCampion This model structure is simplicial. It is transferred from Rezk's model structure which is simplicial. The structure of a simplicial model category can be lifted along the right adjoint functor. The non-existence of a simplicial enrichment is for the category of simplicially enriched categories. Having a space of objects make things a lot easier.

@skd Such a map is called THH-étale, In particular étale maps (of connective rings) are THH-étale.

@SaalHardali I like the following paper by Boavida and Weiss : arxiv.org/abs/1202.1305. You can view embedding calculus as non-monoidal and contravariant version of factorization topology. The poin tis you try to approximate the functor Emb(-,M) by restricting to dijoint union of disks and then right Kan extending it. The main result in this field is that if the codimension is at least 3, this right Kan extension is equivalent to the embedding space.

Should I expect Motive(X)/G to be equivalent to Motive(X/G) ?

However I don't see why they should be equivalent if we only sheafify for the Nisnevich topology.

Say I have a smooth scheme X over a filed with an action of a finite group G such that the quotient X/G exists in the category of smooth schemes. I would like to know how this quotient is related to the (homotopy) quotient taken in the category of motives. The map X-->X/G is an étale cover so the presheaf represented by X/G and the quotient by G of the presheaf represented by X become equivalent after etale sheafification.

in order to prove that K(1)\otimes HQ=0 it suffices to observe that the homotopy groups of that spectrum are both a Q-vector space and an F_p vector space

in general the smah product of two different Morava K-theories K(n)\otimes K(m) is contractible

yes this is not true, the Eilenberg-MacLane spectrum HQ is a counter example: we have K(1)_*HQ=0

@DenisNardin In the simplicial model categories setting you have Theorem 7.10 in this paper : arxiv.org/abs/1410.5675 by Pavlov and Scholbach. In the dg-setting you have this paper : arxiv.org/abs/1311.4130 by Hinich.

Also I gotta go catch a train sorry :)

I don't see how to do this at that level of generality.

If you don't have a natural transformation between F and G but can only construct this map F(I)-->G(I) for this explicit cospan, this might be difficult.

@BrunoStonek The argument you give indeed shows that F(P) and G(P) are connected by a zig-zag of weak equivalences. Wether or not there is an actual map F(P)-->G(P) depends on which point-set level model of the homotopy pushout you use.

@JoeBerner Yes there are two model catgory construction. One due to Fabien Morel "Ensembles profinis simpliciaux et interprétation géométrique du foncteur T" and one due to Isaksen "Completion of pro-spaces". There is also an infinity-categorical version due to Lurie. Together with Ilan Barnea and Yonatan Harpaz, we have written a paper (geoffroy.horel.org/Pro-categories.pdf) in which we compare these three constructions.

yes I mean commutative algebras over HQ

What I mean by "M is a model for X" where M is a model category and X is an infinity category is that when you do the infinty-categorical localization of M at its weak equivalences, you get an \infinity-category that is equivalent to X

Then since both the model categories of CDGAs over Q and of HQ-commutative algebras are combinatorial there must exist a zig-zag of Quillen equivalences (although there is no method for producing it concretely).

@BrunoStonek This proposition is not quite giving what you want. You would need to also prove that the model category of commutative HQ-module is a model for the \infty category of commutative algebras over Q (this is true and I think is proved for symmetric spectra in Pavlov and Scholbach's papers on operads in symmetric spectra).

yes I agree

does this sound correct ?

but since they are both local they are equivaelnt

then THH^R(T) is L-locally equivalent to T

so if T is obtained from R by a Bousfield loc L that is smashing

taking X=S^1 and using your notation you get the equation $THH^T(A)=THH^R(A)otimes_{THH^R(T)}T$

In general they are related by equation 8 of arxiv.org/pdf/math/0306243v1.pdf

@BrunoStonek If R\to T is aTHH etale extension they are the same. The THH etale condition means that THH^R(T) is equivalent to T

@BrunoStonek This is not necessary for the first part of the proposition (see e.g. Hovey's book) but this is necessary for the second part to even make sense.

@BrunoStonek Yes I meant \vee. If E is smashing then E-local objects are stable under tensor product and homotopy colimits. Using the cyclic bar construction, you immediatly see that THH(A) is E-local

@BrunoStonek @BrunoStonek I don' t think so unless the $E$-localization functor is smashing. For instance, if $E_1$ is Morava E-theory of height 1 (i.e. p-completed K-theory), $ THH(E_1)\simeq E_1\wedge \Sigma(E_1)_{\mathbb{Q}}$ is not $K(1)$-local.

@JonBeardsley F_2 is the free group on two generators $\widehat{F}_2$ is its profinite completion

@BrunoStonek You can describe it by the universal property it satisfies. Namely $R[x^{-1}]$ is the initial commutative algebra under $R$ such that the element $x$ is sent to an invertible element.

@BrunoStonek you can define this telescope but the map are not maps of commutative algebras, so it is not clear a priori that the result should have a commutative algebra structure. It turns out that it does in an essentially unique way but this is a non-trivial theorem.

I can't seem to find this in HTT

Does anyone know of a reference for the analogous statement with C an infty-category ?

If C is a small category and A is a small category without loops, then Pro(C^A) is equivalent to Pro(C)^A.

@QiaochuYuan @QiaochuYuan. Maybe what you are looking for is this preprint of mine geoffroy.horel.org/Operads,%20modules%20and%20TFT.pdf. I am in the process of rewriting it to use a more \infty-categorical language. The answer to your question is in Section 3.

so maybe the problem is with this characterization of rigs

then what I have said is fine

do you mean unital monoid ?

there is also the problem of units

maybe

the categorical fact I have said is correct. This is a very easy exercise

so I guess the name you are looking for is commutative algebra

and any object in a category with coproducts is uniquely a monoid with respect to the symmetric monoidal structure given by the coproduct