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Effective Kan Fibrations in Simplicial Sets / by Benno van den Berg, Eric Faber.
- Format:
- Book
- Author/Creator:
- van den Berg, Benno., Author.
- Faber, Eric., Author.
- Series:
- Mathematics and Statistics (SpringerNature-11649)
- Lecture Notes in Mathematics, 1617-9692 ; 2321
- Language:
- English
- Subjects (All):
- Algebra, Homological.
- Logic, Symbolic and mathematical.
- Category Theory, Homological Algebra.
- Mathematical Logic and Foundations.
- Local Subjects:
- Category Theory, Homological Algebra.
- Mathematical Logic and Foundations.
- Physical Description:
- 1 online resource (X, 230 pages) : 1 illustrations
- Edition:
- 1st edition 2022.
- Contained In:
- Springer Nature eBook
- Place of Publication:
- Cham : Springer International Publishing : Imprint: Springer, 2022.
- System Details:
- text file PDF
- Summary:
- 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.
- Other Format:
- Printed edition:
- ISBN:
- 978-3-031-18900-5
- 9783031189005
- Access Restriction:
- Restricted for use by site license.
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