TITLE: Fixed points in category theory
QUESTION [7 upvotes]: 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!

REPLY [10 votes]: 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