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Effective kan fibrations in simplicial sets

Record Type:	Electronic resources : Monograph/item
Title/Author:	Effective kan fibrations in simplicial setsby Benno van den Berg, Eric Faber.
Author:	Berg, Benno van den.
other author:	Faber, Eric.
Published:	Cham :Springer International Publishing :2022.
Description:	x, 230 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Homotopy theory.
Online resource:	https://doi.org/10.1007/978-3-031-18900-5
ISBN:	9783031189005$q(electronic bk.)

Effective kan fibrations in simplicial sets
Berg, Benno van den.

Effective kan fibrations in simplicial sets[electronic resource] /by Benno van den Berg, Eric Faber. - Cham :Springer International Publishing :2022. - x, 230 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v. 23211617-9692 ;. - Lecture notes in mathematics ;2035..

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.

ISBN: 9783031189005$q(electronic bk.)

Standard No.: 10.1007/978-3-031-18900-5doiSubjects--Topical Terms:

209299
Homotopy theory.

LC Class. No.: QA612.7

Dewey Class. No.: 514.24

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520 $a 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.
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