eventual image

Eventual image


The eventual image of an endomorphism f:AAf:A\to A is the intersection n=0 f (n)(A)\bigcap_{n=0}^{\infty} f^{(n)}(A) of the images of its iterates?. This makes sense in many different categories.


On the category of finite sets, the operation assigning to each endomorphism its eventual image is a dinatural transformation.

This example can also be viewed in terms of the trace of FinSet, defined as a:FinSethom(a,a)\int^{a: FinSet}\; \hom(a, a). Indeed, the value of fhom(a,a)f \in \hom(a, a) under the canonical map hom(a,a) ahom(a,a)\hom(a, a) \to \int^a \; \hom(a, a) is the same as that of the restriction f|:Evim(f)Evim(f)f|: Evim(f) \to Evim(f) to its eventual image, which is a permutation, and the value may be regarded as the conjugacy class of that permutation.