2023/10/05 by Ezra Schoen, Schoen, Ezra, Jade Master +3
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic in Computer Science (cs.LO)
paper · doi:10.48550/arxiv.2310.03445
openalex publication_date 2023/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.