Category theory MOC

Slice category

Let 𝖢 be a category and 𝐶 ∈Ob⁡𝖢. The slice category 𝖢/𝐶 is defined as follows:¹ cat

• 𝑓 ∈Ob⁡(𝖢/𝐶) for 𝑋 ∈Ob⁡𝖢 and 𝑓 ∈𝖢(𝑋,𝐶).
• 𝑎 ∈𝖢/𝐶((𝑋,𝑓),(𝑋′,𝑓′)) is a morphism 𝑎 ∈𝖢(𝑋,𝑋′) such that the following diagram commutes:

Typically the objects are referred to by the morphism (e.g. 𝑓) only. A slice category is a special case of a Comma category.

Properties

• There exists a functor 𝑈 :𝖢/𝐶 →𝖢 that forgets the base object.

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Footnotes

1. 2010. Category theory, p. 16 ↩