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?

Thank you for your time!

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 $pi_{n+1}(X)<pi_{n+1}(Xtimes Y)$ and $pi_{k+1}(Y)<pi_{k+1}(Xtimes Y)$).

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 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 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, 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?