Equivalence of categories

Equivalence of categories is a weakening of isomorphism of categories which is not evil. Equivalent categories are “almost” the same in that their categorical properties coïncide.¹ We say a functor $𝐹 : 𝖣 \to 𝐶$ is an (adjoint) equivalence iff there is an adjunction $𝐹 ⊣ 𝐹 - 1$ whose unit and coünit are natural isomorphisms, #m/def/cat thus

𝜂 : 1 ≅ 𝐹 - 1 𝐹, 𝜖 : 𝐹 𝐹 - 1 ≅ 1 .

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.

Notes on the definition

As higher-dimensional structures, equivalences of categories suffers from the same problem as homotopy equivalence and typal equivalence. If we naïvely define $𝐹 : 𝖣 \to 𝖢$ being an equivalence as there existing an $𝐹 - 1 : 𝖢 \to 𝖣$ and natural isomorphisms

𝜂 : 1 ≅ 𝐹 - 1 𝐹, 𝜖 : 𝐹 𝐹 - 1 ≅ 1;

i.e. without the adjointness condition, then $𝜂$ and $𝜖$ are not uniquely determined. However, these data can always be improved to a full adjoint equivalence.

Results

Assuming choice, categories are equivalent iff they have isomorphic skeleta

#state/tidy | #lang/en | #SemBr

This is somewhat circular, since what we typically mean by “categorical property” is a property invariant under equivalence of categories. ↩

Equivalence of categories

Notes on the definition

Results

Footnotes