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?



Cellularity of anodyne extensions?
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 $D$.




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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.


