Spahn mapping simplex (Rev #1)

In the theory of cartesian fibrations of simplicial sets cartesian fibrations $X\to \Delta^n$ over a simplex play an important role since an arbitrary morphism $X\to S$ is a cartesian fibration iff $X\times_S X$ is a cartesian fibration.

$X\to \Delta^n$ is by the $(\infty,1)$-Grothendieck construction equivalently a functor $\Delta^n\to (\infty,1)Cat$; i.e. a composable sequence of $(\infty,1)$-categories and functors $\phi:A^0\leftarrow\dots\leftarrow A^n$.

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 $\sigma^L\to A^{f(j)}$.

• 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 $\phi$.