S1 - Index Of Arrow

Premise: After being presumed dead for five years, Oliver Queen returns to Starling City with a list of names of corrupt people his father left behind. Using archery, combat skills, and a hooded persona, he wages a one-man war against crime as the Arrow while confronting secrets from his past.
Tone & style: Gritty, grounded superhero drama that blends present-day vigilante missions with flashbacks revealing Oliver’s island survival and training.
Key themes: Redemption, consequences of vigilantism, family loyalty, corruption, identity.

Goal: Formulate a clear, general definition of "index of Arrow s1", place it in context (category theory, directed graphs, dynamical systems, or index theory), and develop computation and applications.
Assumption: "Arrow s1" denotes a distinguished morphism s1: A → A (endomorphism) or a distinguished directed edge in a given structure. We adopt the endomorphism interpretation first, then note alternatives.