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.
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