The eventual image (Q6593824)

Any endomorphism in a suitably complete category \(\mathcal{C}\)\ gives rise to an automorphism in \(\mathcal{C}\)\ in two dual universal ways. Let \(X^{\circlearrowright f}\)\ be an endomorphism in \(\mathcal{C}\). Following from the existence of Kan extensions,\N\N\begin{itemize}\N\item[(1)] there is an automorphism \(L^{\circlearrowright u}\)\ in \(\mathcal{C}\)\ together with a map \(L^{\circlearrowright u}\rightarrow X^{\circlearrowright f}\)\ that is terminal among all maps from an automorphism to \(X^{\circlearrowright f}\), and dually\N\N\item[(2)] there is an automorphism \(M^{\circlearrowright v}\)\ in \(\mathcal{C}\)\ together with a map \(X^{\circlearrowright f}\rightarrow M^{\circlearrowright v}\)\ that is initial as such,\N\end{itemize}\N\NIt is observed that in categories whose objects are sufficiently finite in nature, these two dual universal constuctions coincide, called the \textit{eventual image} of \(f\).\N\NThis paper establishes that in any category with a factorization system abiding by certain axioms, the eventual image has two dual universal properties. The author characterizes the eventual image as a terminal coalgebra.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] gives the definitions.\N\N\item[\S 3] gives some general results such as the claim that\N\[\N\mathrm{im}^{\infty}\left( f^{n}\right) =\mathrm{im}^{\infty}\left( f\right) \text{ and }\left( f^{n}\right) ^{\infty}=f^{\infty}\text{ for every }n\geq1\text{.}\N\]\N\N\item[\S 4] establishes the main result claiming that if \(\mathcal{C}\)\ admits a factorization system of finite type, then every endomorphism in \(\mathcal{C}\)\ has eventual image duality.\N\N\item[\S 5] establishes the theorem claiming that in a category with a factorization system of finite type, the eventual image is the terminal coalgebra for an endofunctor \(A\mapsto fA\).\N\N\item[\S 6] analyzes \ the eventual image in the category of finite sets.\N\N\item[\S 7] analyzes \ the eventual image in the category of finite-dimensional vector spaces.\N\N\item[\S 8] analyzes \ the eventual image in the category of compact metric spaces.\N\N\item[\S 9] gathers further examples of the eventual image, showing, for instance, that in a Cauchy-complete category whose hom-sets are finite, every endomorphism has eventual image duality.\N\end{itemize}