Definition:Isomorphism of Categories
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Definition
Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be a functor.
Then $F$ is an isomorphism (of categories) if and only if there exists a functor $G: \mathbf C \to \mathbf D$ such that:
- $G F: \mathbf C \to \mathbf C$ is the identity functor $I_{\mathbf C}$
- $F G: \mathbf D \to \mathbf D$ is the identity functor $I_{\mathbf D}$
Isomorphic Categories
Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories.
Then $\mathbf C$ and $\mathbf D$ are said to be isomorphic, and we write $\mathbf C \cong \mathbf D$.
Also see
Linguistic Note
The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus isomorphism means equal structure.
Sources
- 1998: Saunders Mac Lane: Categories for the Working Mathematician (2nd ed.): $\S \text{IV}.4$: Equivalence of Categories