# A note on fixed points of endofunctors

## TL;DR

This paper explores fixed points of endofunctors in category theory, introducing three classes, analyzing their properties, and relating fixed points to sheaf cohomology in categories with pretopology.

## Contribution

It introduces three classes of fixed points for endofunctors and characterizes them using sheaf cohomology in categories with pretopology.

## Key findings

- Existence of induced structures on fixed point categories
- Characterization of fixed points via sheaf cohomology
- Analysis of fixed points in categories with pretopology

## Abstract

In this note, we deal with the fixed points of an endofunctor $F: \mathcal{C} \longrightarrow \mathcal{C}$. Three classes of fixed points are introduced, and the case when $F$ is an endomorphism of a category with pretopology is investigated. We show existence of induced structures on the category of fixed points, and, when a pretopology is defined, give a characterization of fixed points in terms of sheaf cohomology on $\mathcal{C}$.