Effective Kan Fibrations in Simplicial Sets

Benno van den Berg, Eric Faber (Autoren)

Buch | Softcover

X, 230 Seiten

2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-18899-2 (ISBN)

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This book introduces the notion of an effective Kan fibration, a new mathematical structure which can be used to study simplicial homotopy theory. The main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. Here it is revealed that fundamental properties of ordinary Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, a constructive (explicit) proof is given that effective Kan fibrations are stable under push forward, or fibred exponentials. Further, it is shown that effective Kan fibrations are local, or completely determined by their fibres above representables, and the maps which can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly, both notions still describe the same homotopy theory. These new results solve an open problem in homotopy type theory and provide the first step toward giving a constructive account of Voevodsky's model of univalent type theory in simplicial sets.

- 1. Introduction. - Part I - Types from Moore Paths. - 2. Preliminaries. - 3. An Algebraic Weak Factorisation System from a Dominance. - 4. An Algebraic Weak Factorisation System from a Moore Structure. - 5. The Frobenius Construction. - 6. Mould Squares and Effective Fibrations. - 7. -Types. - Part II Simplicial Sets. - 8. Effective Trivial Kan Fibrations in Simplicial Sets. - 9. Simplicial Sets as a Symmetric Moore Category. - 10. Hyperdeformation Retracts in Simplicial Sets. - 11. Mould Squares in Simplicial Sets. - 12. Horn Squares. - 13. Conclusion.

Erscheinungsdatum	11.12.2022
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	X, 230 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	379 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Constructive Mathematics • homotopy theory • Homotopy Type theory • Kan Complexes • Simplicial sets
ISBN-10	3-031-18899-3 / 3031188993
ISBN-13	978-3-031-18899-2 / 9783031188992
Zustand	Neuware