Given a left fibration $X\to S$, we want to show that there is a map $S\to \mathcal{S}$ such that $X\to S$ is induced from the *universal* left fibration $\mathcal{S}_{1/}\to \mathcal{S}$ in the sense that there is a homotopy pullback diagram (??in the Joyal model structure on $\mathbf{S}$??)

$\array{
X & \rightarrow & \mathcal{S}_{1/} \\
\downarrow & & \downarrow \\
S & \rightarrow & \mathcal{S}
}$

Furthermore, we want to show that the classifying map $S\to \mathcal{S}$ is unique up to equivalence.

Observe firstly that there are 1-1 correspondences between commutative diagrams

$\array{
X & \to & \mathcal{S}_{1/} \\
\downarrow & & \downarrow \\
S & \to & \mathcal{S}
}$

in $\mathbf{S}$, maps

$1\star X\cup_X S \to \mathcal{S}$

in $\mathbf{S}$, and finally simplicial functors

$\mathfrak{C}(1\star X\cup_X S)\to \mathbf{Kan}.$

Let $v$ denote the cone point in $1\star X$ and let $\mathcal{M}$ denote a fibrant replacement of $\mathfrak{C}(1\star X\cup_X S)$.

Then we obtain a simplicial functor

$\mathfrak{C}(1\star X\cup_X S)\to \mathbf{Kan}$

by the formula

$\Map_{\mathcal{M}}(v,-)\colon \mathfrak{C}(1\star X\cup_X S)\to \mathbf{Kan}.$

Revised on September 24, 2013 at 10:42:03
by
Danny Stevenson