#
The eventual image

##
Tom Leinster

In a category with enough limits and colimits, one can form the universal
automorphism on an endomorphism in two dual senses. Sometimes these dual
constructions coincide, including in the categories of finite sets,
finite-dimensional vector spaces, and compact metric spaces. There,
beginning with an endomorphism f, there is a doubly-universal
automorphism on f whose underlying object is the eventual image
∩_{n >= 0} im(f^n). Our main theorem unifies these examples,
stating that in any category with a factorization system satisfying certain
axioms, the eventual image has two dual universal properties. A further
theorem characterizes the eventual image as a terminal coalgebra. In all,
nine characterizations of the eventual image are given, valid at different
levels of generality.

Keywords:
dynamical system, factorization system, coalgebra, metric space

2020 MSC:
18A32, 18A40, 18F99, 51F99

*Theory and Applications of Categories,*
Vol. 42, 2024,
No. 9, pp 180-221.

Published 2024-07-24.

http://www.tac.mta.ca/tac/volumes/42/9/42-09.pdf

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