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Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)

If not, is there a known counterexample?

Similarly, does the same hold for inner anodyne extensions and inner horn inclusions?

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    $\begingroup$ When you take the geometric realization of an anodyne extension of simplicial sets, you get what is called a simple homotopy equivalence (more or less by definition). There is an invariant of homotopy equivalences called Whitehead torsion which must vanish for a map $f$ to be homotopic to a simple homotopy equivalence. I'm guessing that you can get a counterexample by taking a homotopy equivalence with nontrivial Whitehead torsion, finding a simplicial model for it, and taking the mapping cylinder to get an inclusion whose realization is a homotopy equivalence not homotopic to a simple one. $\endgroup$
    – Tyler Lawson
    Commented Oct 24, 2018 at 10:56
  • $\begingroup$ @Tyler Did you mean that the geometric realization of a pushout of a horn inclusion is always simple, rather than all anodynes? Otherwise, this would not produce a counterexample. $\endgroup$
    – Harry Gindi
    Commented Oct 24, 2018 at 11:01
  • $\begingroup$ @HarryGindi you wrongly use the feature "@", by not letting the viewer complete the name. Tyler won't be aware of your post in the top part of the window unless he rereads your question. $\endgroup$
    – Philippe Gaucher
    Commented Oct 24, 2018 at 12:49
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    $\begingroup$ @PhilippeGaucher It doesn't appear on the mobile version of the website. $\endgroup$
    – Harry Gindi
    Commented Oct 24, 2018 at 12:54
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    $\begingroup$ @HarryGindi yes, that's correct, I meant the geometric realization of an iterated pushout along horn inclusions. Sorry for the confusion. $\endgroup$
    – Tyler Lawson
    Commented Oct 24, 2018 at 16:08

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I believe the following seem to be a very simple counterexample without it being related to Whitehead obstruction as suggested by Tyler Lawson.

Consider the simplicial sets $D$ freely generated by:

  • a $0$-cell $x$.

  • a $1$-cell $t:x \rightarrow x$

  • a $2$-cell $\gamma$ such that $d_0 \gamma = d_1 \gamma = d_2 \gamma = t$

It is contractible: indeed it is connected, Its $\pi_1$ is generated by $t$ subject to the relation $(\gamma) : t^2=t$ hence is trivial. Its homology is easy to compute and is trivial.

The inclusion of $\Delta[0]$ into $D$ is hence a cofibration and a weak equivalence, i.e. a trivial cofibration.

$D$ has only three non-degenerate cells, and taking a pushout by a Horn inclusion $\Lambda^k[n] \hookrightarrow \Delta[n]$ always add exactly two non-degenerate cells of dimension $(n-1)$ and $(n)$.

So if $\Delta[0] \hookrightarrow D$ were a pushout of Horn inclusion it would have to be a pushout by $\Lambda^i[2] \hookrightarrow \Delta[2]$.

Now for each $i$ there is only one maps $\Lambda^i[2] \rightarrow \Delta[0]$ along which to take the pushout and none of them gives $D$.

Here is an attempt to get a counter example in the case of the Joyal model structure.

I'm constructing $D'$ by adding to $D$ the following cells:

  • a $1$-cell $u:x \rightarrow x$.
  • a $2$-cell $\theta$ such that $d_0 \theta =t$, $d_2 \theta = u$ and $d_1 \theta$ is the unique degenerate $1$-cell.

$D'$ is contractible for the Joyal model structure: its homotopy category is freely generated by an idempotent with a left inverse, hence is trivial. Hence $D'$ is an $\infty$-groupoids with trivial $\pi_1$ and the computation of its homology shows that it is contractible.

If it were obtained from $\Delta[0]$ as pushout of Inner Horn inclusion the argument of number of non-degenerate cells shows that it has to be a pushout of to copies of $\Lambda^1[2] \rightarrow \Delta[2]$.

But if that were the case, $t$ would have to be added as pushout of $\Lambda^1[2] \rightarrow \Delta[2]$ at some points, in particular $t$ would appears as $d_1 y$ for $y$ a non-degenerate two cell, with $d_0 y$ and $d_2 y$ not equal to $t$. but there are only two non-degenerate two cells and none of them has this property.

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  • $\begingroup$ Where is the homotopy between the degenerate 1-simplex and t? Otherwise, it's a model for a monoid with a nontrivial idempotent. $\endgroup$
    – Harry Gindi
    Commented Oct 24, 2018 at 15:35
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    $\begingroup$ I was talking of the Kan-Quillen model structure. Not the Joyal model structure. It also work for Lurie/Verity model structure on marked simplicial set (by marking $t$) , but indeed not for the Joyal model structure. $\endgroup$
    – Simon Henry
    Commented Oct 24, 2018 at 15:37
  • $\begingroup$ Ah, I see. I'm going to accept this answer, but if you can think of a counterexample for inner anodynes, please add it too, as a challenge =)! $\endgroup$
    – Harry Gindi
    Commented Oct 24, 2018 at 15:38
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    $\begingroup$ @HarryGindi : Done ! well I think... $\endgroup$
    – Simon Henry
    Commented Oct 24, 2018 at 16:02

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