Effective Kan Fibrations in Simplicial Sets

This book introduces the concept of effective Kan fibrations, which are maps of simplicial sets with a structured collection of lifts that satisfy certain non-trivial properties. It extends fundamental properties of ordinary Kan fibrations to explicit constructions on effective Kan fibrations, proving that they are stable under push forward, fibred exponentials, local, and completely determined by their fibers above representables. This solves an open problem in homotopy type theory and provides a step toward a constructive account of Voevodskys model of univalent type theory in simplicial sets.

Format: Paperback / softback
Length: 230 pages
Publication date: 10 December 2022
Publisher: Springer International Publishing AG

This book presents a groundbreaking concept known as an effective Kan fibration, a novel mathematical framework that empowers the study of simplicial homotopy theory. The primary objective is to enhance the applicability of simplicial homotopy theory to homotopy type theory. Effective Kan fibrations are intricate maps of simplicial sets adorned with a carefully selected set of lifts that adhere to specific non-trivial conditions. Here, it is unveiled that fundamental characteristics of ordinary Kan fibrations can be extended to explicit constructions on effective Kan fibrations. Specifically, a constructive (explicit) proof is provided that effective Kan fibrations remain stable under push forward, or fibered exponentials. Moreover, it is demonstrated that effective Kan fibrations are local, or entirely determined by their fibers above representables, and the maps that can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Consequently, implicitly, both notions continue to describe the same homotopy theory. These groundbreaking findings resolve an open problem in homotopy type theory and mark the initial step toward providing a constructive interpretation of Voevodskys model of univalent type theory in simplicial sets.

Weight: 379g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031188992
Edition number: 1st ed. 2022