Canonical self-indexing

Let $𝖢$ be a category. The canonical self-indexing $𝖢 / ⇥ 𝖢$ is defined so that #m/def/cat/dis

an object over $𝑎 \in 𝖢 0$ is a pair $(𝑥, 𝑓)$ where $𝑥 \in 𝖢 0$ and $𝑓 : 𝑥 \to 𝑎$ ;
for $ℎ : 𝑎 \to 𝑏$ in $𝖢$ , a morphism $ℎ' : (𝑥, 𝑓) \to (𝑦, 𝑔)$ is a commuting square, i.e. a morphism $ℎ' : 𝑥 \to 𝑦$ in $𝖢$ such that $𝑔 ℎ' = ℎ 𝑓$ .

The “display map” arrows in a diagram in $𝖢 / ⇥ 𝖢$ literally correspond to morphisms in $𝖢$ . The fibre categories of $𝖢$ are equivalent to slice categories. Dualizing, we can form the cocanonical self-indexing $𝖢\ ⇥𝖢$ , whose fibres are equivalent to coslice categories.

As a fibration

A fundamental property of the canonical self-indexing is that it is a Cartesian fibration iff $𝖢$ has all pullbacks. This follows from the fact that the diagram in $𝖢 / ⇥ 𝖢$ says the same thing as the diagram in $𝖢$ on the right

so $𝑓$ and $𝜑'$ are precisely the pullbacks of $𝑔$ and $𝜑$ respectively.

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