[001X] Construction 2.1·a (The canonical self-indexing).

Let $B$ be an ordinary category; there is a canonical displayed category $\SelfIx{B}$ over $B$ given fiberwise by the slices of $B$.

  1. For $x\in B$, we define $\SelfIx{B}\Sub{x}$ to be the collection $\Sl{B}{x}$ of pairs $(\bar{x}\in B,p\Sub{x}:\bar{x}\to x)$.
  2. For $f : x\to y\in B$, we define $\SelfIx{B}\Sub{f}$ to be the collection of commuting squares in the following configuration: