Hi guys, I have a question. Prove or disproof the statement:


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Any inner horn $\Lambda[n,k],0< k< n $ admits a filtration $\mathbf{n}<\cdots<\Lambda[n,k]$, such that each step is filling an inner horn.


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where $\mathbf{n}$ is the simplicial set $0\to 1 \cdots \to n$ (this notation is bad), its 1-cells are $(i,i+1)$, do not include $(i,j), i+1< j$. For those who does not familiar with simplicial set, one could just think of $\Lambda[n,k]$ as an indexed geometrical horn. So basically this is a combinatary question. For $n=2,3$, this is apparent, for $n=4$ one can still do it by hand. I guest it is true for higher $n$. 


Thanks for your reading.



Decomposition of inner horns
Joyal and Tierney also prove (extending the result of Cordier and Porter) a large class of statements of this form (including, in addition to this fact, that the inclusion of the spine into a simplex (that is, the inclusion of $n$ into $\Delta^n$ in your notation) is inner anodyne). This is in the paper by Joyal and Tierney comparing quasicategories and complete Segal spaces. I think the title is something like "quasicategories vs segal spaces". They do this by using a formalism of "generalized horns".


Similar results were derived by Rezk in his paper on $\Theta\_n$-spaces using a "cover" formalism, which are, for all intents and purposes, generalized spines. This approach has the advantage of admitting a homotopy-theoretic rather than a combinatorial proof. However, while covers do generate the class of inner anodynes, they do not, unfortunately, include the inner horns. 


A Rezk cover of a simplex $\Delta[n]$ is a subpresheaf containing the spine $\operatorname{Sp}[n]$ having the right lifting property with respect to all "cospine" inclusions $\Delta[1]=\Delta[\{0,m\}]\hookrightarrow \Delta[m]$. In particular, all covers are stable under pullbacks along "sequential" maps (maps that induce maps on spines).


