12
$\begingroup$

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 $\Lambda^n_k$.

$\endgroup$
1
  • 8
    $\begingroup$ This is a pretty natural question, and I thought about it for a while (but never found an answer either way). Good luck! $\endgroup$ Nov 7, 2015 at 18:35

1 Answer 1

7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.