Equivalence of Categories

Isomorphism of categories demands a bijection on objects and almost never holds; equivalence asks only for full, faithful and essentially surjective — and choosing a basis is exactly one.

The idea

Equivalence of categories is the working notion of two categories being the same. The strictest candidate — functors $F \colon \mathcal{A} \to \mathcal{B}$ and $G \colon \mathcal{B} \to \mathcal{A}$ with $G \circ F = 1_{\mathcal{A}}$ and $F \circ G = 1_{\mathcal{B}}$ — is isomorphism of categories, and it is too strict to be useful: the equalities force $F$ to be a bijection on objects, so a category is not even permitted to carry a redundant second copy of an object it already has.

The definition weakens equality to natural isomorphism.

Definition (Equivalence of categories).

An equivalence between $\mathcal{A}$ and $\mathcal{B}$ is a pair of functors $F \colon \mathcal{A} \to \mathcal{B}$ and $G \colon \mathcal{B} \to \mathcal{A}$ together with natural isomorphisms $\eta \colon 1_{\mathcal{A}} \Rightarrow G \circ F, \qquad \varepsilon \colon F \circ G \Rightarrow 1_{\mathcal{B}}.$ The categories are then equivalent, written $\mathcal{A} \simeq \mathcal{B}$.

Producing all of that data is laborious, and there is a test that mentions only $F$. Call $F$ faithful when it is injective on each collection of arrows $A \to A'$, full when it is surjective on each, and essentially surjective when every object of $\mathcal{B}$ is isomorphic to $F(A)$ for some $A$ — isomorphic, not equal.

Theorem.

A functor $F \colon \mathcal{A} \to \mathcal{B}$ is an equivalence if and only if it is full, faithful and essentially surjective on objects.

The word isomorphic gives the definition its reach. Finite-dimensional real vector spaces are far too numerous to be matched one for one with the natural numbers, yet sending $n$ to $\mathbb{R}^{n}$ and a matrix to the linear map it defines is an equivalence, because every such space is isomorphic to some $\mathbb{R}^{n}$. The equivalence amounts to choosing a basis for each space.

Ways to work on it

• Walkthrough. Why isomorphism of categories is too strict, the definition by natural isomorphisms, the full-faithful-essentially-surjective test, and matrices against vector spaces.
• Proof. An equivalence is exactly a full, faithful, essentially surjective functor — including the construction of the inverse, and the choice it depends on.
• Practice. Test functors for fullness, faithfulness and essential surjectivity, and compare small categories up to equivalence.
• Hardest. Decide which small categories are equivalent, including pairs that no object or arrow count can separate.

Not sure where to start? Take the ten-question placement test.