Equivalence of Categories

Idea

An equivalence of categories relates two categories $\mathcal{C}$ and $\mathcal{D}$ in a manner that is invertible up to isomorphism, thus is weaker than a full isomorphism of categories. It can be thought of as a quasi-inverse in the category of categories.

Definition

An equivalence of categories is a functor

$$F:\mathcal{C}\Rightarrow \mathcal{D}$$

that preserves all categorical structure up to isomorphism, and is invertible up to natural isomorphism rather than on the nose.

Concretely, $F$ is an equivalence if there exists a functor

$$G:\mathcal{D}\Rightarrow \mathcal{C}$$

together with natural isomorphisms

$$\eta :{\mathrm{Id}}_{\mathcal{C}}\cong G\circ F\epsilon :F\circ G\cong {\mathrm{Id}}_{\mathcal{D}}$$

So $G$ behaves like an inverse to $F$, but only up to coherent isomorphism.

Practical Criterion

A functor $F:\mathcal{C}\to \mathcal{D}$ is an equivalence iff it is:

• fully faithful: for all objects $X,Y$ in $\mathcal{C}$, the map $${\mathrm{Hom}}_{\mathcal{C}}(X,Y)\to {\mathrm{Hom}}_{\mathcal{D}}(FX,FY)$$ is a bijection
• essentially surjective on objects: every object of $\mathcal{D}$ is isomorphic to some $FX$

This is usually the most useful characterization.

Example

Any category $\mathcal{C}$ is equivalent to its skeletons, assuming sufficient choice:

$$\mathcal{C}\simeq \mathrm{sk}(\mathcal{C}).$$

References

riehl2016-category-theory