Isomorphism (category theory)

N.B. this node concerns itself with isomorphisms, a special type of morphism in a category. For isomorphism between categories, see Isomorphism of categories.

Definition

A morphism $f:A\to B$ in an arbitrary category is called an isomorphismadj. isic. if there is an inverse morphism $g:B\to A$ satisfying fg = \id{A} ParseError: Unexpected end of input in a macro argument, expected '}' at end of input: fg = \id{A} and gf = \id{B} ParseError: Unexpected end of input in a macro argument, expected '}' at end of input: gf = \id{B}.

The existence of an isomorphism between two objects $A$ and $B$ gives rise to a relation denoted $A\cong B$. In English, one may say that $A$ is isomorphic to $B$.

Properties

• Isomorphism is an equivalence relation; it is reflexive, transitive, and symmetric.

• Every identity morphism is an isomorphism and its own inverse.

Examples

• In $𝐒𝐞𝐭$, the isomorphisms are precisely the bijective functions.

• If every morphism in a monoid category is an isomorphism, then the monoid is a group.

Automorphism

An isomorphism that is also an endomorphism is called an automorphism. One special example of automorphism is an identity arrow, which is called a trivial automorphism.

References

• [cite:@riehl2017category]