## hopf algebras – Do chains send homotopy inverse limits of spaces to homotopy inverse limits of \$E_infty\$-coalgebras?

Let $$X_bullet := … X_2 to X_1$$ be a tower of connected and simple spaces
with the following properties:

The induced tower $$H_ast(X_bullet; mathbb{F}_p)$$ of graded $$mathbb{F}_p$$-vector spaces
is Mittag-Leffler and lifts to a tower of graded abelian hopf algebras over $$mathbb{F}_p$$.
The induced tower $$pi_1(X_bullet)$$ is Mittag-Leffler.

By a theorem of Goerss the canonical morphism $$H_ast(holim X_bullet; mathbb{F}_p)to lim H_ast(X_bullet; mathbb{F}_p)$$ is an isomorphism, where the limit on the right hand side is taken in the category of graded abelian hopf algebras over $$mathbb{F}_p$$.

The limit in the category of graded abelian hopf algebras over $$mathbb{F}_p$$ forgets to the limit in the category of graded cocommutative coalgebras over $$mathbb{F}_p$$ but does generally not forget to the limit in graded $$mathbb{F}_p$$-vector spaces.

Is there a similar result on the chain level?

Is it true under the assumptions on $$X_bullet$$ that the canonical morphism $$C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ is a quasi-isomorphism, where
$$C_ast(-; mathbb{F}_p)$$ are chains with $$mathbb{F}_p$$-coefficients, and the homotopy limit on the right hand side is taken in the $$infty$$-category of $$E_infty$$-coalgebras over $$mathbb{F}_p$$?

This would follow of course from Goerss theorem if homology $$H_ast$$ would send
the homotopy limit in $$E_infty$$-coalgebras (over $$mathbb{F}_p$$) of the tower $$C_ast(X_bullet; mathbb{F}_p)$$ to the limit in graded cocommutative coalgebras over $$mathbb{F}_p$$.

Does one know such a result?

Can one say more if one additionally assumes that the tower $$X_bullet := … X_2 to X_1$$
of spaces refines to a tower of grouplike $$E_infty$$-spaces?

Under this assumption the induced tower $$C_ast(X_bullet; mathbb{F}_p)$$
is a tower of $$E_infty$$-hopf algebras over $$mathbb{F}_p$$, i.e. abelian group objects (in the derived sense) in the $$infty$$-category of $$E_infty$$-coalgebras over $$mathbb{F}_p.$$

Therefore by Goerss theorem the canonical morphism $$C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ would be a quasi-isomorphism if homology $$H_ast$$ would send
the homotopy limit in $$E_infty$$-hopf algebras of the tower $$C_ast(X_bullet; mathbb{F}_p)$$ to the limit in graded abelian hopf algebras over $$mathbb{F}_p$$.

Does one know such a result?

## homological algebra – Comparing notions of \$A_{infty}\$ homotopy (in char 0): Markl’s definition versus “Sullivan homotopy”

Suppose that we have $$A_{infty}$$ algebras $$A,B$$ (over a field of characteristic $$0$$), with $$A_{infty}$$ maps $$f,g: A rightarrow B$$. In the paper https://arxiv.org/abs/math/0401007 Markl defines a notion of $$A_{infty}$$ homotopy which looks like a some kind of elaborate chain homotopy condition (top of page 4, item $$(6)$$). At least over characteristic $$0$$, there is a competiting definition of a homotopy, defined as an $$A_{infty}$$ morphism $$H: A rightarrow B otimes Omega^{bullet}_{(0,1)}$$ such that $$H|_{t = 0} = f$$ and $$H|_{t =1} = g$$.

Supposedly, in this context, the two definitions are equivalent. How can do I go from Markl’s definition to the other definition?

## at.algebraic topology – Whitehead product and a homotopy group of a wedge sum

Note : this is a crosspost from the Mathematics StackExchange, as suggested by this meta post.

Let $$X$$ be an $$n$$-connected ($$ngeqslant1$$) CW-complex and $$Y$$ be a $$k$$-connected ($$kgeqslant1$$) CW-complex. My goal is to prove the following isomorphism : $$pi_{n+k+1}(Xvee Y)congpi_{n+k+1}(X)opluspi_{n+k+1}(Y)oplus(pi_{n+1}(X),pi_{k+1}(Y)),$$
with $$(;cdot;,;cdot;)$$ denoting the Whitehead product (here, it is understood that we take the whitehead product of the subgroups $$pi_{n+1}(X) and $$pi_{k+1}(Y)).

So far, I have done the following. (Do let me know if I have done any mistake !)

We can always assume, up to a homotopy equivalence, by the hypothesis on $$X$$ and $$Y$$, that their respective $$n$$ and $$k$$ skeletons are of the following form : $$text{Sk}_nX={ast}qquadtext{and}qquadtext{Sk}_kY={ast}.$$
In particular, $$X$$ and $$Y$$ only have cells in dimensions $$geqslant n+1$$ and $$geqslant k+1$$ respectively. Therefore, the product $$Xtimes Y$$ has only cells starting in dimension $$n+1$$ or $$k+1$$, accordingly to which one is the smallest, and that cells in dimensions $$leqslant n+k+1$$ come from cells of either $$X$$ or $$Y$$, but not both. Therefore, we get : $$text{Sk}_{n+k+1}(Xtimes Y)subset Xvee Y,$$
and thus the pair $$(Xtimes Y,Xvee Y)$$ is $$(n+k+1)$$-connected.

I then tried using a part of the exact sequence of the pair :

$$dotslongrightarrowpi_{n+k+2}(Xtimes Y,Xvee Y)overset{partial_ast}{longrightarrow}pi_{n+k+1}(Xvee Y)overset{imath_ast}{longrightarrow}pi_{n+k+1}(Xtimes Y)overset{text{rel}_ast}{longrightarrow}pi_{n+k+1}(Xtimes Y,Xvee Y)longrightarrowdots$$

We can use the $$(n+k+1)$$-connectedness of the pair to re-write the sequence as :

$$dotslongrightarrowpi_{n+k+2}(Xtimes Y,Xvee Y)overset{partial_ast}{longrightarrow}pi_{n+k+1}(Xvee Y)overset{k}{longrightarrow}pi_{n+k+1}(X)opluspi_{n+k+1}(Y)overset{text{rel}_ast}{longrightarrow}0,$$

with $$k$$ being given by the composite of $$imath_ast$$ and of the isomorphism $$pi_bullet(Xtimes Y)congpi_bullet(X)opluspi_bullet(Y)$$.

Now, the sequence splits at $$pi_{n+k+1}(Xvee Y)$$, since we have $$pcircimath=text{id}$$ and $$qcircimath=text{id}$$ in : $$Xvee Yoverset{imath}{longrightarrow}Xtimes Yoverset{p}{longrightarrow}Xsubset Xvee Yqquadtext{and}qquad Xvee Yoverset{imath}{longrightarrow}Xtimes Yoverset{q}{longrightarrow}Ysubset Xvee Y,$$

by functoriality and by using that $$pi_bullet$$ sends products to products. We shall denote as $$p_astoplus q_ast:pi_{n+k+1}(X)opluspi_{n+k+1}(Y)topi_{n+k+1}(Xvee Y)$$ the splitting retraction. Therefore, by an algebraic lemma (not exactly the Splitting lemma, but something rather similar), we obtain : $$pi_{n+k+1}(Xvee Y)congtext{Im}(p_astoplus q_ast)oplusker(k).$$

Now, I recognized that $$text{Im}(p_astoplus q_ast)congpi_{n+k+1}(X)opluspi_{n+k+1}(Y)$$ by construction, so I am left with computing $$ker(k)$$. And here, I am completely stuck… How to recognize the Whitehead product as the kernel I am missing ?

## at.algebraic topology – Spaces homotopy equivalent over the topologist’s sine curve

Consider $$T=left{ left( x, sin tfrac{1}{x} right ) : x in (0,1) right} cup {(0,0)}subset mathbb{R}^2$$
with its subspace topology.

Denote $$p_0=(0, 0)in T, p_1=(1, sin 1)in T$$.

A TSC-homotopy between continuous maps $$f, g:Xto Y$$ is a continuous map $$H:Xtimes Tto Y$$ such that $$H(x, p_0)=f, H(x, p_1)=g$$.

What can be said about pairs of spaces $$X, Y$$ such that there exist continuous maps $$f:Xto Y, g:Yto X$$ and TSC-homotopies $$fcirc gsim mathrm{id}_Y, gcirc fsimmathrm{id}_X$$?

## Does the homotopy category of a model category detect weak equivalences?

Let $$mathcal{C}$$ be a model category. Suppose that a morphism $$f:xto y$$ in $$mathcal{C}$$ induces an isomorphism in the homotopy category $$textbf{Ho}(mathcal{C})$$. Is it necessarily true that $$f$$ is a weak equivalence in $$mathcal{C}$$?

If not, are there nice sufficient conditions for this to hold? For instance, does it hold for simplicial model categories?

## homological algebra – homotopy coherent G-action on tensor product of complexes

Let $$G$$ be a discrete group and $$k$$ a field. Suppose $$C_1$$ and $$C_2$$ are complexes over $$k$$ with homotopy coherent actions of $$G$$ in the sense of Cordier (I’ve been reading https://arxiv.org/pdf/1801.07404.pdf as a reference). $$C_1otimes_k C_2$$ should inherit a homotopy coherent $$G$$ action which induces the diagonal $$G$$-action on $$H_*(C_1 otimes_k C_2) cong H_*(C_1)otimes_k H_*(C_2).$$

How do I explicitly construct this homotopy coherent $$G$$-action on $$C_1 otimes_k C_2$$? I’m interested in some explicit universal formulas (or a method for generating such formulas) in terms of the $$G$$-actions on $$C_1,C_2.$$ It is elementary to see how to do this “by hand” at the level of “2-simplices” (i.e. right down a homotopy between the composition of diagonal actions of $$g,h$$ and the diagonal action of $$gh$$). I could probably do something similar for “3-simplices,” but I feel like this should be standard for people who know more homotopical algebra.

## What fails in constructing a homotopy category out of candidate triangles in a triangulated category?

Following Neeman’s article “New axioms for triangulated categories”, for a triangulated category $$mathscr T$$ let $$CT(mathscr T)$$ denote the category of candidate triangles, i.e. diagrams
$$begin{equation}Xoverset fto Yoverset gto Zoverset hto Sigma Xquad (*)end{equation}$$
such that $$gf=0$$, $$hg=0$$ and $$(Sigma f)h=0$$, with morphisms being commutative diagrams between such triangles.
We can define homotopy of maps between candidate triangles to be chain homotopy and there is an automorphism $$tildeSigmacolon CT(mathscr T)to CT(mathscr T)$$ which takes $$(*)$$ to
$$Yoverset{-g}to Zoverset{-h}to Sigma Xoverset{-Sigma f}to Sigma Y.$$
We can define mapping cones as in a usual chain complex category, and a lot of the usual results hold for this category (e.g. homotopic maps have isomorphic mapping cones).

What I fail to see, is why the mapping cone construction along with $$tildeSigma$$ does not give rise to a triangulation of $$CT(mathscr T)$$?

## higher category theory – Counterexamples concerning \$infty\$-topoi with infinite homotopy dimension

In “Higher Topos Theory”, Lurie introduces three different notions of dimension for an $$infty$$-topos $$mathcal{X}$$, namely:

• Homotopy dimension (henceforth h.dim.), which is $$leq n$$ if $$n$$-connective objects admit global sections.
• Local Homotopy dimension $$leq n$$ if there exist objects $${ U_alpha }$$ generating $$mathcal{X}$$ under colimits such that $$mathcal{X}_{/U_alpha}$$ is of h.dim. $$leq n$$.
• Cohomological dimension (coh.dim.) $$leq n$$ if for $$k>n$$ and any abelian group object $$A in operatorname{Disc}(mathcal{X})$$, we have $$operatorname{H}^k(mathcal{X},A) = 0$$.

Corollary 7.2.2.30 shows that if $$n geq 2$$, and $$mathcal{X}$$ is an $$infty$$-topos that has finite h.dim. and coh. dim. $$leq n$$, then it also has h.dim. $$leq n$$. While the converse (h.dim $$leq n$$ then also coh.dim. $$leq n$$) always holds, the extra requirements are definitely necessary for the given proof; and there is even a counterexample given in 7.2.2.31 for an $$infty$$-topos that is of coh.dim. 2, but has infinite h.dim.:

Let $$mathbb{Z}_p$$ be the p-adic integers regarded as a profinite group. The example is constructed by forming an ordinary category $$mathcal{C}$$ of the finite quotients $${ mathbb{Z}_p/{p^n mathbb{Z}_p}}_{n geq 0}$$, equipping it with a Grothendieck topology where any nonempty sieve ist covering, and forming the (evidently 1-localic) $$infty$$-topos $$mathcal{X}=Shv(Nmathcal{C})$$. While I don’t completely understand the p-adic methods used in the proof that this is of infinite homotopy dimension, the gist is the following: An $$infty$$-connective morphism $$alpha$$ in $$mathcal{X}$$ ist constructed and it is shown that $$alpha$$ can’t be an equivalence, so that $$mathcal{X}$$ is not hypercomplete and, due to Corollary 7.2.1.12, can therefore not be of locally finite homotopy dimension. This is where my issue with the proof lies: Locally finite homotopy dimension does not imply finite homotopy dimension, neither the other way around:

• In Post #80 here, Marc Hoyois gives an example of a cohesive (therefore also finite h.dim.) $$infty$$-topos that is not hypercomplete, and can’t be locally of finite h.dim because of this. Further, I was told that sheaves over Spectra, e.g. of $$mathbb{Z}$$, with the étale topology often also are counterexamples of this direction.
• I unfortunately do not know an example of an $$infty$$-topos that is of finite local h.dim. but not of finite h.dim.; I would be happy if anyone could think of one.

It seems to me that this proof in HTT is not complete because of this, and therefore I wanted to ask whether I just didn’t properly understand the argument, a part is missing or if the example maybe doesn’t even work at all.

## ag.algebraic geometry – Freyd non-concreteness in motivic homotopy theory

The following theorem is due to Freyd.

Let $$C$$ be a category of pointed topological spaces including all finite-dimensional CW-complexes. Let $$H:Cto mathrm{Set}$$ be a homotopy invariant functor. There exists $$f:Xto Y$$ such that $$f$$ is not null-homotopic but $$H(f)=H(star)$$ where $$star$$ is null-homotopic.

Is there an analogue of this theorem in motivic homotopy theory?

## transition in homotopy theory

I guess that the following are true; maybe classical? Is there a reference?

1. Let $$X, Y, Z$$ be connected pointed CW-complexes, $$f:Xto Y$$ and $$g:Xto Z$$ pointed maps. Assume that
for every $$kge 1$$, the kernel of $$f_*:pi_k(X)topi_k(Y)$$ is contained in the kernel of $$g_*:pi_k(X)topi_k(Z)$$. Does it follow that there is a pointed map $$h:Yto Z$$ such that $$hcirc f$$ is homotopic to $$g$$?

2. Let $$Diff(M)$$ be the group of the smooth diffeomorphisms of a closed separated manifold $$M$$, with the smooth topology. Does $$Diff(M)$$ have the homotopy type of a numerable CW-complex? Are its integral homology groups numerable?