# Remark 2.4.1.4 Higher Topos Theory

In HTT, given a inner fibration $$p : X \rightarrow S$$ of simplicial, an edge $$f : x \rightarrow y$$ of the simplicial set $$X$$ is said to be a $$p$$-Cartesian if the induced map $$X_{/f} \rightarrow X_{/y} \times_{S_{/p(y)}} S_{/p(f)}$$ is a trivial Kan fibration.

In Remark 2.4.1.4 Lurie says that this definition is equivalent to the following one : in the same setting, $$f$$ is $$p$$-Cartesian if and only if for every $$n \geq 2$$ and every commutative diagram

$$\begin{matrix} \Delta^{\{n-1,n\}} \\ \downarrow \scriptstyle&&\\ \Lambda^n_n&{\to}&X\\ \downarrow &&\downarrow \scriptstyle{p}\\ \Delta^n&{\to}&S\end{matrix}$$ where the composition $$\Delta^{\{n-1,n\}} \rightarrow \Lambda^n_n \rightarrow X$$ is the edge $$f$$ ( I don't know how to make a "bottom right" arrow) there is a map $$h : \Delta^n \rightarrow X$$ rendering the diagram commutative.

I don't see why those definitions are equivalent. For example if we have the first one, given a diagram $$\begin{matrix} \Delta^{\{n-1,n\}} \\ \downarrow\scriptstyle&&\\ \Lambda^n_n &\stackrel{\alpha}{\to}&X\\ \downarrow & &\downarrow \scriptstyle{p}\\ \Delta^n&\stackrel{\beta}{\to}&S\end{matrix}$$

we have a map $$\Delta^{n-2} \rightarrow S_{/p(f)}$$ using $$\beta$$ and the universal property of the slice but I don't see how to get a map from $$\Delta^{n-2}$$ into $$X_{/y}$$ nor from a simplicial subset of $$\Delta^{n-2}$$ into $$X_{/f}$$.

## 1 Answer

Let's see what the data of a diagram

$$\begin{matrix} \partial\Delta^{n-2}&{\to}&X_{/f}\\ \downarrow &&\downarrow\\ \Delta^{n-2}&{\to}&X_{/y}\times_{S_{/p(y)}}S_{p(f)}\end{matrix}$$

translates to. I claim this is formally the same as a diagram

$$\begin{matrix} \partial\Delta^{n-2}\star\Delta^{\{n-1,n\}}\sqcup_{\partial\Delta^{n-2}\star\Delta^{\{n\}}}\Delta^{n-2}\star\Delta^{\{n\}}&{\to}&X\\ \downarrow &&\downarrow\\ \Delta^{n-2}\star\Delta^{\{n-1,n\}}&{\to}&S\end{matrix}$$

and that the data of a lift is the same in both diagrams. This currying-like pattern appears over and over again in HTT - maybe the place where it is spelled out most explicitly is in Remark A.3.1.6. (which is for mapping spaces, but essentially the same thing applies for under- or over-categories.)

Anyways, the left hand arrow coincides with $$\Lambda^n_n\rightarrow\Delta^n$$, proving the claim.