# Extending a left fibration along an inner horn

Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times_{\Delta^n} \Lambda^n_i$?

The only results in that direction that I am aware of are the following two lemmas from Left fibrations and homotopy colimits by Heuts, Moerdijk:

Lemma 7.3. Consider a pullback square of simplicial sets $\require{AMScd}$ \begin{CD} X \times_Y Z @>g>> Z\\ @VVV @VV p V\\ X @>>f> Y \end{CD} in which $f$ is inner anodyne and $p$ is a left fibration. Then $g$ is a trivial cofibration in the Joyal model structure.

Lemma 7.4. Let $0 < k < n$ and let $p : A \rightarrow \Lambda^n_k$ be a left fibration. Then there exists a left fibration $q : B \rightarrow \Delta^n$ and an equivalence \begin{CD} A @>g>> \Lambda^n_k \times_{\Delta^n} B\\ @VpVV @VVV\\ \Lambda^n_k @= \Lambda^n_k \end{CD} in the covariant model structure over $\Lambda^n_k$.

• This is a pretty natural question, and I thought about it for a while (but never found an answer either way). Good luck! – Charles Rezk Nov 7 '15 at 18:35

Yes. This is shown using minimal left fibrations in Cisinski's book Higher categories and homotopical algebra, see the proof of Theorem 5.2.10 therein. This theorem states that the simplicial set $$\mathscr{S}$$, whose $$n$$-simplices are (essentially) the left fibrations over $$\Delta^n$$ with small fibres, is a quasi-category.