Fibrations of simplicial sets

TL;DR

This paper explores various variants of Kan fibrations in simplicial sets, analyzing their combinatorial properties and how they interact with homotopy types, revealing the richness of the fibrations' structure.

Contribution

It introduces and investigates multiple variants of Kan fibrations that satisfy Quillen's axioms, emphasizing their combinatorial finitary nature and interaction with homotopy types.

Findings

Multiple fibrations satisfy Quillen's axioms

Finitary combinatorics underlie these fibrations

Iterates of Kan's Ex functor reveal distinctions among fibrations

Abstract

There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan's Ex functor act on graphs and on nerves of small categories.