Isomorphism

Definition

Let $\mathcal C$ be a category . A morphism $f : X \to Y$ is an isomorphism if there exists a morphism $g : Y \to X$ such that

where $1_X$ and $1_Y$ are the identity morphisms and $\circ$ is composition .

The morphism $g$ is unique if it exists, and is denoted $f^{-1}$. In this case, $X$ and $Y$ are said to be isomorphic (in $\mathcal C$).

Related notions:

• An isomorphism $X\to X$ is an automorphism .
• Every isomorphism is both a monomorphism and an epimorphism .

Examples

1. In $\mathbf{Set}$, the isomorphisms are exactly the bijective functions .
2. In $\mathbf{Grp}$, the isomorphisms are exactly group isomorphisms (bijective homomorphisms).
3. In $R\text{-}\mathbf{Mod}$, the isomorphisms are exactly $R$-linear maps with $R$-linear inverses (invertible module homomorphisms).