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In the theory of cartesian fibrations of simplicial sets cartesian fibrations$X\to \Delta^n$cartesian fibration s over of a simplicial simplex sets play cartesian an fibrations important role since an arbitrary morphism$X\to S{\Delta}^{n}$ X\to S \Delta^n is over a cartesian simplex fibration play iff an important role since an arbitrary morphism$X{\times}_{S}\to \mathrm{XS}$ X\times_S X\to X S is a cartesian fibration iff for all $n$, $X\times_S \Delta^n\to \Delta^n$ is a cartesian fibration.
$X\to \Delta^n$A cartesian fibration is by the $X\to \Delta^n$$(\infty,1)$ is by the -Grothendieck construction equivalently a functor $(\infty,1)$$\Delta^n\to (\infty,1)Cat$-Grothendieck construction equivalently a functor ; i.e. a composable sequence of $\Delta^n\to (\infty,1)Cat$$(\infty,1)$; i.e. a composable sequence of -categories and functors $(\infty,1)$$\phi:A^0\leftarrow\dots\leftarrow A^n$-categories and functors $\phi:A^0\leftarrow\dots\leftarrow A^n$.
The mapping simplex $M(\phi)$ of $\phi$ is defined by:
The mapping simplex $M(\phi)$ of $\phi$ is defined by:
For a nonempty finite finite linear order $L$ with greatest element $j$, a map $\Delta^L\to M(\phi)$ consists of a order preserving map $f:L\to [n]$ and a morphism $\sigma:\Delta^L\to A^{f(j)}$.
Given two such linear orders $L$ and $L^\prime$ with greatest elements $j$ resp. $j^\prime$ there is a natural map $M(\phi)(\Delta^{L^\prime})\to M(\phi)(\Delta^{L})$ sending $(f,\sigma)$ to $(f\circ p, e\circ \sigma)$, where $e:A^{f(j^\prime)}\to A^{f(p(j))}$ is obtained by $\phi$.
There is a natural map $h:M(\phi)\to \Delta^n$ (take $J=m$, then the Yoneda lemma gives a map $\Delta^m\to \Delta^n$).
An edge $e$ of $M(\phi)$ is defined by a pair of integers $0\le i\le j\le n$ and an edge $e^\prime\in A^j$. $M(\phi)$ becomes a marked simplicial set $(M(\phi), E)$ by marking those edges for which $e^\prime$ is degenerated.
For a nonempty finite finite linear order $L$ with greatest element $j$, a map $\Delta^L\to M(\phi)$ consists of a order preserving map $f:L\to [n]$ and a morphism $\sigma^L\to A^{f(j)}$.
Given two such linear orders $L$ and $L^\prime$ with greatest elements $j$ resp. $j^\prime$ there is a natural map $M(\phi)(\Delta^{L^\prime})\to M(\phi)(\Delta^{L})$ sending $(f,\sigma)$ to $(f\circ p, e\circ \sigma)$, where $e:A^{f(j^\prime)}\to A^{f(p(j))}$ is obtained by $\phi$.
Let $p:X\to \Delta^n$ be a cartesian fibration, let $\phi:A^0\leftarrow\dots\leftarrow A^n$ be a composable sequence of $(\infty,1)$-categories and functors. Then A map $q:M(\phi)\to X$ is called a quasi-equivalence if it satisfies:
(1) The map $h$ commutes with $p$ and $q$.
(2) $q$ sends marked edges of $M(\phi)$ to $p$-cartesian ones.
(3) For every $0\le i\le n$, the induced map $A^i\to p^{-1}\{ i \}$ is a categorical equivalence?.
Let $p:X\to \Delta^n$ be a cartesian fibration.
(1) There exists a composable sequence of $(\infty,1)$-categories and functors $\phi:A^0\leftarrow\dots\leftarrow A^n$ and a quasi-equivalence $q:M(\phi)\to X$.
(2) If $\phi:A^0\leftarrow\dots\leftarrow A^n$ is a composable sequence and $q:M(\phi)\to X$ a quasi-equivalence. Then for any map $T\to \Delta^n$, the induced map
is a categorical equivalence?.