Equivalence of categories

Let $\mathcal C,\mathcal D$ be categories .

Definition

A functor $F:\mathcal C\to\mathcal D$ is an equivalence of categories if there exists a functor $G:\mathcal D\to\mathcal C$ and natural isomorphisms

In this case $G$ is called a quasi-inverse (or weak inverse) of $F$, and we say $\mathcal C$ and $\mathcal D$ are equivalent.

Equivalently, $F$ is an equivalence iff:

1. (Fully faithful) for all objects $X,Y\in\mathcal C$, the induced map on hom-sets $\mathrm{Hom}_{\mathcal C}(X,Y)\longrightarrow \mathrm{Hom}_{\mathcal D}(F(X),F(Y))$ is a bijection, and
2. (Essentially surjective) every object $D\in\mathcal D$ is isomorphic to some $F(C)$.

(These conditions make precise that $F$ preserves and reflects all morphism data, and hits every object up to isomorphism.)

Examples

1. Finite sets vs. standard finite sets (FinSet).
   Let $\mathbf{FinSet}$ be the category of finite sets. Let $\mathbf{FinStd}$ be the full subcategory whose objects are the standard sets $[n]=\{1,2,\dots,n\}$ (including $[0]=\varnothing$).
   The inclusion $\mathbf{FinStd}\hookrightarrow \mathbf{FinSet}$ is an equivalence: every finite set is bijective to exactly one $[n]$ up to isomorphism, and morphisms are just set functions.

2. Finite-dimensional vector spaces vs. matrices.
   Let $\mathbf{FinVect}_k$ be finite-dimensional $k$-vector spaces. Let $\mathbf{Mat}_k$ be the category with:

   • objects natural numbers $n$,
   • morphisms $n\to m$ given by $m\times n$ matrices over $k$,
   • composition given by matrix multiplication.
     The functor $F:\mathbf{Mat}_k\to \mathbf{FinVect}_k$ sending $n\mapsto k^n$ and a matrix to the corresponding linear map is an equivalence (choosing a basis identifies any finite-dimensional space with some $k^n$).

3. A category is equivalent to its skeleton.
   A skeleton of $\mathcal C$ is a full full subcategory containing exactly one object from each isomorphism class of objects in $\mathcal C$.
   The inclusion of a skeleton $\mathrm{Sk}(\mathcal C)\hookrightarrow \mathcal C$ is an equivalence: it is fully faithful (being full) and essentially surjective by construction.