Slice category

Let $𝖢$ be a category and $𝐶 \in O b 𝖢$ . The slice category $𝖢 / 𝐶$ is defined as follows:¹ #m/def/cat

an object in $𝖢 / 𝐶$ is a pair $(𝑋, 𝑓)$ where $𝑋 \in 𝖢 0$ and $𝑓 : 𝑋 \to 𝐶$ ;
a morphism $𝑎 : (𝑋, 𝑓) \to (𝑋', 𝑓')$ is a commuting triangle, i.e. a morphism $𝑎 : 𝑋 \to 𝑋'$ in $𝖢$ such that $𝑓' 𝑎 = 𝑓$ .

Typically the objects are referred to by the morphism (e.g. $𝑓$ ) only. Dualizing, we can form the coslice category $𝖢\𝐶$ . Both are naturally viewed as special cases of the more general Comma category, or as the fibres of the canonical and cocanonical self indexing respectively.

Properties

There exists a functor $𝑈 : 𝖢 / 𝐶 \to 𝖢$ that forgets the base object.

#state/develop | #lang/en | #SemBr

2010. Category theory, p. 16 ↩

Slice category

Properties

Footnotes