| Description |
1 online resource (x, 230 pages) : illustrations |
| Series |
Lecture notes in mathematics ; volume 2321 |
|
Lecture notes in mathematics (Springer-Verlag) ; 2321.
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| Contents |
4 An Algebraic Weak Factorisation System from a Moore Structure -- 4.1 Defining the Algebraic Weak Factorisation System -- 4.1.1 Functorial Factorisation -- 4.1.2 The Comonad -- 4.1.3 The Monad -- 4.1.4 The Distributive Law -- 4.2 Hyperdeformation Retracts -- 4.2.1 Hyperdeformation Retracts are Coalgebras -- 4.2.2 Hyperdeformation Retracts are Bifibred -- 4.3 Naive Fibrations -- 5 The Frobenius Construction -- 5.1 Naive Left Fibrations -- 5.2 The Frobenius Construction -- 6 Mould Squares and Effective Fibrations -- 6.1 A New Notion of Fibred Structure -- 6.2 Effective Fibrations |
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6.2.1 Effective Trivial Fibrations -- 6.2.2 Right and Left Fibrations -- 7 -Types -- Part II Simplicial Sets -- 8 Effective Trivial Kan Fibrations in Simplicial Sets -- 8.1 Effective Cofibrations -- 8.2 Effective Trivial Kan Fibrations -- 8.3 Local Character and Classical Correctness -- 9 Simplicial Sets as a Symmetric Moore Category -- 9.1 Polynomial Yoga -- 9.2 A Simplicial Poset of Traversals -- 9.3 Simplicial Moore Paths -- 9.4 Geometric Realization of a Traversal -- 10 Hyperdeformation Retracts in Simplicial Sets -- 10.1 Hyperdeformation Retracts Are Effective Cofibrations |
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10.2 Hyperdeformation Retracts as Internal Presheaves -- 10.3 A Small Double Category of Hyperdeformation Retracts -- 10.4 Naive Kan Fibrations in Simplicial Sets -- 11 Mould Squares in Simplicial Sets -- 11.1 Small Mould Squares -- 11.2 Effective Kan Fibrations in Terms of ̀̀Filling'' -- 12 Horn Squares -- 12.1 Effective Kan Fibrations in Terms of Horn Squares -- 12.2 Local Character and Classical Correctness -- 13 Conclusion -- 13.1 Properties of Effective Kan Fibrations -- 13.2 Directions for Future Research -- A Axioms -- A.1 Moore Structure -- A.2 Dominance -- B Cubical Sets |
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C Degenerate Horn Fillers Are Unique -- D Uniform Kan Fibrations -- References -- Index |
| 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 Voevodskys model of univalent type theory in simplicial sets |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Homotopy theory.
|
|
Homotopía
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Homotopy theory
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Teoria de l'homotopia.
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| Genre/Form |
Llibres electrònics.
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| Form |
Electronic book
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| Author |
Faber, Eric, author
|
| ISBN |
9783031189005 |
|
3031189000 |
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