Displayed category of subobjects

Let $𝖢$ be a category. The displayed category $𝖲 𝗎 𝖻 ⇥ 𝖢$ is defined so that

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

This is a restriction of the canonical self-indexing. $𝖲 𝗎 𝖻$ is so named since it may be viewed as a displayed category of subobjects. Its total category is called the category of monomorphisms.

Properties

$𝖲 𝗎 𝖻$ is thinly displayed.

Proof

Let $𝑓 : 𝑎 \to 𝑏$ , $𝑖 𝑥 : 𝑥 ↣ 𝑎$ , and $𝑖 𝑦 : 𝑦 ↣ 𝑏$ , and suppose $𝑓', 𝑓 ″$ lie over $𝑓$ . Then $𝑖 𝑦 𝑓' = 𝑓 𝑖 𝑥 = 𝑖 𝑦 𝑓 ″$ so by the definition of a monomorphism $𝑓' = 𝑓 ″$ , proving ^P1.

Displayed category of subobjects

Properties

See also