Equivalence

Formal Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be Categories, Let $\mathcal{A}\simeq \mathcal{B}\equiv$ ”$\mathcal{A}$ and $\mathcal{B}$ are equivalent”, Let $F_{\mathcal{A}\mathcal{B}}$ be a Functor

$$\begin{array}{c}\exists F_{\mathcal{A}\mathcal{B}}:\mathcal{A}\to \mathcal{B},F_{\mathcal{B}\mathcal{A}}:\mathcal{B}\to \mathcal{A}\\ F_{\mathcal{B}\mathcal{A}}\circ F_{\mathcal{A}\mathcal{B}}\cong {\text{id}}_{\mathcal{A}}\\ F_{\mathcal{A}\mathcal{B}}\circ F_{\mathcal{B}\mathcal{A}}\cong {\text{id}}_{\mathcal{B}}\\ ⟺\\ \mathcal{A}\simeq \mathcal{B}\end{array}$$

• This definition is very similar to that of Isomorphism of Categories