Fixed points in category theory

Question

Is there a usual way to express the concept of fixed points in category theory? What would I say to express that a morphism has a fixed point? Thank you!

Answer 1

A fixed point is a solution to the equation $x = f(x)$. The usual way to encode equations is as equalizers.

Given an object $A$ and an endomorphism $f : A \to A$, we can define "the subobject $B \mapsto A$ fixed by $f$" to be the equalizer of $f$ and the identity morphism. This subobject has the universal property that if $g : C \to A$ is any morphism with $fg = g$, then $g$ factors uniquely through $B$. (i.e. we can factor $g$ as $C \to B \to A$)

Answer 2

It is more interesting to ask whether an endofunctor has a fixed point. One distinguishes between least fixed points and largest fixed points. For example, the unit intervall is the largest bipointed topological space with a "self-similarity" $I = I \vee I$, and the set of natural numbers is the least set with $\mathbb{N} = 1 + \mathbb{N}$, the set of binary trees is the least set with $T=1+T^2$. For details, references and more examples, see:

• Tom Leinster, A universal Banach space
• Andreas Blass, Seven Trees in One
• nlab, Initial algebra of an endofunctor
• nlab, Terminal coalgebra of an endofunctor
• nlab, Coalgebra for the real interval