Isomorphism

Formal Definition

An Isomorphism is a kind of Morphism between objects in a Category that is Invertible.

Let $f_{AB}$ be a Morphism and $A,B$ be objects

$$\begin{array}{c}\exists f_{AB}:A\to B,f_{AB}^{-1}:B\to A\\ f_{AB}^{-1}\circ f_{AB}={\text{id}}_A\wedge f_{AB}\circ f_{AB}^{-1}={\text{id}}_B\\ ⟺\\ A\cong B\end{array}$$

Isomorphism Facts

• Isomorphisms are unique
• If there exists an isomorphism between objects, then those objects are isomorphic with each other
• If $A$ and $B$ are isomorphic, then $A\cong B$
• When objects are isomorphic, we can say that they have the same internal structure, without having to actually look inside the objects
• For example, as Vector Spaces: $\mathbb{C}\cong {\mathbb{R}}^2$ (not as Algebraic Fields though)

Isomorphism of Categories

Categories contain Objects, and Objects can be anything, including Categories!

• Functors connect Categories
• Morphisms connect Objects
• Our Objects are Categories
• Thus, our morphisms are functors The Objects of a Categories being Categories themselves, is just a special case of Category. Therefore, the same general Formal Definition applies. However, it can also be reexpressed in terms of functors, where $\mathcal{A},\mathcal{B}$ are categories, and $F_{\mathcal{A}\mathcal{B}}$ is a functor.

$$\begin{array}{c}\exists F_{\mathcal{A}\mathcal{B}}:\mathcal{A}\to \mathcal{B},F_{\mathcal{A}\mathcal{B}}^{-1}:\mathcal{B}\to \mathcal{A}\\ F_{\mathcal{A}\mathcal{B}}^{-1}\circ F_{\mathcal{A}\mathcal{B}}={\text{id}}_{\mathcal{A}}\\ F_{\mathcal{A}\mathcal{B}}\circ F_{\mathcal{A}\mathcal{B}}^{-1}={\text{id}}_{\mathcal{B}}\\ ⟺\\ \mathcal{A}\cong \mathcal{B}\end{array}$$

Size Intuition

• When there is a notion of size, this invertibility can be thought of as a Bijective Morphism between a pair of Objects; a relationship where one object can be converted and restored to and from another object.
• A Bijective Homomorphism.

Consider the Arrows to and from category $A$ to $B$, that are Bijective because $A$ and $B$ have the size

$$\begin{array}{c}{f}_{AB}:A\to B\\ f_{BA}:B\to A\\ f_{AB}^{-1}=f_{BA}\\ f_{AB}^{-1}\circ f_{AB}={\text{id}}_A\\ f_{AB}\circ f_{AB}^{-1}={\text{id}}_B\end{array}$$