Relative fixed points of functors

TL;DR

This paper explores the concept of relative fixed points of functors, establishing their properties, characterizations, and existence conditions within locally presentable categories, with applications to polynomial functors and examples like the Sierpinski carpet.

Contribution

It introduces the notion of relative fixed points for functors, characterizes them via (co)equalizers, and provides conditions for their existence and preservation, extending the theory of fixed points in category theory.

Findings

Relative fixed points form adjoint pairs between F-coalgebras and F-algebras.

Characterization of relative fixed points as (co)equalizers of free (co)monads.

Existence conditions for relative fixed points in locally presentable categories.

Abstract

We show how the relatively initial or relatively terminal fixed points for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find sufficient conditions for existence of this adjunction. We show that relative fixed points may be characterized as (co)equalizers of the free (co)monad on $F$. In particular, when $F$ is a polynomial functor on $\mathsf{Set}$ the relative fixed points are a quotient or subset of the free term algebra or the cofree term coalgebra. We give examples of the relative fixed points for polynomial functors and an example which is the Sierpinski carpet. Lastly, we prove a general preservation result for relative fixed points.