6.1.3. Example: The dual self-indexing [0019]

Dually to the canonical self-indexing (Construction 2.1·a [001X]), every category $B$ can also be displayed over itself via its coslices $\CoSl{x}{B}$.

[002Z] Construction 6.1.3·a.

Let $B$ be a category. Define the displayed category $\overline{B}$ over $B$ as follows:

1. For $x\in B$, define $\overline{B}\Sub{x}$ as the collection of pairs $(\bar{x}\in B,p\Sub{x}:x\to\bar{x})$.
2. For $f : x\to y\in B$, define $\overline{B}\Sub{f}$ to be the collection of commuting squares in the following configuration:
[001M] Exercise 6.1.3·b.

Prove that $\overline{B}$ is a cocartesian fibration if and only if $B$ has pushouts.

[001N] Exercise 6.1.3·c.

Prove that the total category (Section 3.1 [000A]) of $\overline{B}$ is the arrow category $B^{\to}$, and the projection is the domain functor.

[0030] Exercise 6.1.3·d.

Prove that $\overline{B}$ is a cartesian fibration for any category $B$.