Definition:Equivalence of Categories
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Definition
Let $\mathbf C$ and $\mathbf D$ be metacategories.
An equivalence of $\mathbf C$ and $\mathbf D$ comprises:
- Functors $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf C$
- Natural isomorphisms $\alpha: G F \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf C}$ and $\beta: F G \overset{\sim}{\longrightarrow} \operatorname{id}_{\mathbf D}$
Equivalent Categories
Let there exist an equivalence of categories between $\mathbf C$ and $\mathbf D$.
Then $\mathbf C$ and $\mathbf D$ are said to be equivalent, denoted $\mathbf C \simeq \mathbf D$.
Also defined as
Some sources call $F$ an equivalence if there exist $G, \alpha$ and $\beta$ as above.
Also see
Sources
- 1998: Saunders Mac Lane: Categories for the Working Mathematician (2nd ed.): $\S \text{IV}.4$: Equivalence of Categories