The eventual image

TL;DR

This paper unifies various dual universal properties of the eventual image in categories with certain factorization systems, providing nine characterizations and linking it to terminal coalgebras.

Contribution

It introduces a general framework for the eventual image with dual universal properties and characterizes it as a terminal coalgebra across broad categorical contexts.

Findings

Unifies dual universal properties of the eventual image.

Provides nine different characterizations of the eventual image.

Links the eventual image to terminal coalgebras.

Abstract

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, as 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 $\bigcap_n \mathrm{im}(f^n)$. 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.