Definition:Isomorphism (Category Theory)
This page is about Isomorphism in the context of Category Theory. For other uses, see Isomorphism.
Definition
Let $\mathbf C$ be a category, and let $X, Y$ be objects of $\mathbf C$.
A morphism $f: X \to Y$ is an isomorphism if and only if there exists a morphism $g: Y \to X$ such that:
- $g \circ f = I_X$
- $f \circ g = I_Y$
where $I_X$ denotes the identity morphism on $X$.
It can be seen that this is equivalent to $g$ being both a retraction and a section of $f$.
Inverse Morphism
A morphism $g: Y \to X$ is said to be an inverse (morphism) for $f$ if and only if:
- $g \circ f = I_X$
- $f \circ g = I_Y$
where $I_X$ denotes the identity morphism on $X$.
Also known as
Some authors, to avoid tedium, speak simply of an iso.
Furthermore, in place of the consistent phrasing "$f$ is an iso" they will generally prefer the shorter "$f$ is iso".
Linguistic Note
The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus isomorphism means equal structure.
Also see
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.5$: Definition $1.3$