Let X be a simplicial space. We have an obvious forgetful functor F: sTop -> sSet, given by postcomposition with the forgetful functor Top -> Set. It seems like there should be a natural map |X| -> |F(X)| of geometric realizations. What can be said about this map (if it exists)? Is it a weak homotopy equivalence?

Basically, what does geometric realization remember about a simplicial space beyond the geometric realization of its underlying simplicial set?

@ReubenStern definitely not a weak homotopy equivalence: for example, let X be the constant simplicial space valued in S^1; its geometric realization is S^1. But F(X) is the constant simplicial set valued in the set S^1, which is some uncountable set, so its geometric realization is that set, which is an uncountable discrete space.

Geometric realization glues the simplices together. But for simplicial spaces, you have the option to glue things other than just the standard n-simplex, and remember their topology; in simplicial sets, you build the topology out of the simplices, and each simplex is pretty boring homotopically

The two do agree if the simplicial space you've started with is levelwise discrete. But yeah, in general it seems like you're losing way too much information by forgetting from spaces to sets?

I guess another example is that you can build the classifying space of a discrete group G as the geometric realization of the bar construction B(*, G, *), a simplicial set. (This is in May's Concise Course, though maybe not in quite these words). But if G is a topological group (in geometry, the classifying spaces one cares about are usually for compact Lie groups), this construction doesn't quite work, and you have to take the bar construction B(*, G, *) as a simplicial space.

Can anyone say WHY I can think of A_\infty algebras (in the sense of algebra, not topology, say) as formal non-commutative dg-manifolds?

(a la Kontsevich-Soibelman)

@JonathanBeardsley that was my thought, too. Might there be some way to quantify the "information lost" through this forgetting?

@ReubenStern i'm not sure. i imagine one could approach such a statement by saying "these are the spaces for which the two results agree"

i keep thinking of complete segal spaces... i don't really know. but just thinking about some kind of condition that says "up to homotopy your simplicial space doesn't have any more info its underlying simplicial set"

but this seems like, basically, you'd be asking for all of the spaces in your simplicial space to just be discrete up to homotopy.

Right. I'll keep thinking! Not even sure yet I have a well-defined natural transformation |-| => |F(-)|...

but... it's hard for me to think globally about an entire simplicial object at once. i often think about the levels and don't think about the maps.

it might be worth thinking about the simplicial spaces you get by taking "topological simplices" of a space X

i.e. the sTop version of Sing(-)

I think |Sing(X)| should get you something equivalent to X (I think this is actually some kind of replacement functor in a model category

@ReubenStern There isn't one in that direction simply because there is no continuous function X --> (set of points of X with discrete topology) natural in X --and you'd need this for the case of constant simplicial objects. The identity in the other direction is continuous, but of course it is basically never a homotopy equivalence. (The whole simplicial business is irrelevant here, since all these issues arise already for constant simplicial spaces.)

@skd very cool! I'm definitely interested in the Gross-Hopkins and Scholze-Weinstein stuff.

@JonathanBeardsley regarding A_\infty algebras. Are you looking for something more than just the fact that an A_infty algebra structure on a graded vector space V is equivalent to a deg 1 codifferential on T(V[-1]) i.e. the cofree coassociative coalgebra generated by V[-1]?

ps: hi btw. First time caller. Long time lurker.

@YuriSulyma me too; if you find out more about this, let me know!

one thing that's somewhat related which i find very interesting: a lot of things (classically) have analogues in adic geometry which are well-behaved, but don't play nicely with homotopy theory. the lubin-tate tower is an example, as are modular curves with p-power level structures

idk whether this is hinting at some sort of general principle

@ChrisR oh no, I mean, that answers my question for sure!

@SaulGlasman @TylerLawson @DylanWilson Thanks for the discussion/reference!

Here is a question about Goodwillie calculus I cannot seem to find a good answer to. Is it proven somewhere in the literature, or an easy consequence of some more general theorem?

The question is: If I have two homogeneous functors $F$ and $G$ (from spectra to spectra) of degrees $n$ and $m$ resp., can one calculate the natural transformations from $F$ to $G$?

I guess I should be more specific. I am interested in the natural transformations $(-)^{\otimes n}\rightarrow (-)^{\otimes m}_{h\Sigma_m}$. I expect there to be no transformations unless $n=m$, and in the case $n=m$ I expect a unique transformation up to contractible choice.

Hi Espen, this space of natural transformations is generally only contractible if m < n. If m=n, then natural transformations are given by \Sigma_n-equivariant maps between the coefficient spectra (the derivatives) of the functors involved. All of this is in Calculus III.

For n<m things are more complicated, since this will involve the "dual derivatives" (in the sense of McCarthy) of the functor (-)^{\otimes m}_{h\Sigma_m}. For example, the space of natural transformations from id (the case n=1) to (-)^{\otimes 2}_{h\Sigma_2} is equivalent to \Omega^\infty\Sigma^{-1} S^{t\Sigma_2}, i.e., the zeroth space of a shift of the \Sigma_2 Tate construction of the sphere spectrum (which is the 2-completion of the sphere spectrum).

Thanks, Gijs! Do you have a reference for McCarthy's paper?

@GijsHeuts Thanks!

A probably naïve question: let $E$ be a $G$-spectrum, do we know $\pi_n^G(E)$ if we know $pi_n(E^G)$？ If we don't, what can we say about $\pi_n^G(E)$?

aren't those two groups equal by definition?

Maybe he meant fixed points of htpy groups for the first one? In that case they're often different, and knowledge of either does not determine the other

naïve questions about equivariant homotopy theory are welcome as long as they're genuine :)

@SaulGlasman true but not by definition, S^n has trivial G-action so it has to be mapped into the G-fixed point of E.

But the definition here cannot be generalized to the RO(G)-graded homotopy.

@DylanWilson the case I am thinking about is what will happen if we consider the RO(G)-graded HFPSS, where the E_2 page is H^s(G;\pi_0(S^V\wedge E)) and it converges to (according to Hill-Meier 15 paper) \pi_{V-s}^G(F(EG_+,E)). When $V$ is a trivial representation it is just the HFPSS.

But when $V$ is not trivial and assume for simplicity that $E$ is the Eilenberg-MacLane spectrum $H\underline{Z}$, are the nonvanishing terms in the E_2 page only the ones whose second grading in RO(G) are the $V$s with virtual degree 0?

@ArunDebray that is a genuine statement:)