Isomorphism - Imported Knowlpedia Content

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

g\circ f = 1_X \quad\text{and}\quad f\circ g = 1_Y,

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:

Examples

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