Category theory MOC

Equivalence of categories

Equivalence of categories is a weakening of isomorphism of categories which in the majority of cases is actually more appropriate. Equivalent categories are “almost” the same in that their categorical properties coïncide. An equivalence of categories 𝖢,𝖣 is a pair 𝐹 :𝖢 ⇆𝖣 :𝐺 of functors such that

𝐺𝐹≃1𝖢,𝐹𝐺≃1𝖣

where ( ≃) denotes natural isomorphism. This is reminiscent of Homotopy equivalence. We also see equivalence of categories is a special case of an adjunction of functors for which the unit and coünit are isomorphisms.

Results

• Categories are equivalent iff they have isomorphic skeleta

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