Construction (The Covariant Transport Functor). Let $q: X \rightarrow S$ be a left fibration of simplicial sets, and let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction Combining Proposition, Proposition, and Example, we deduce that there is a unique map of simplicial sets $T: S \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Kan}} )$ with the following properties:

  • For each vertex $s \in S$, we have $T(s) = X_{s}$.

  • For each edge $e: s \rightarrow s'$ of the simplicial set $S$, we have $T(e) = [ e_{!} ]$. Here $e_{!}: X_{s} \rightarrow X_{s'}$ denotes a morphism of Kan complexes given by covariant transport along $e$, and $[e_{!}]$ denotes its homotopy class.

The morphism $T$ determines a functor from the homotopy category $\mathrm{h} \mathit{S}$ (Notation to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, which we will also denote by $T: \mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$. We will refer to $T$ as the covariant transport functor.