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.
Author(s): Benno van den Berg, Eric Faber
Series: Lecture Notes in Mathematics, 2321
Publisher: Springer
Year: 2023
Language: English
Pages: 229
City: Cham
Preface
Related Work
Acknowledgements
Contents
1 Introduction
1.1 Fibrations as Structure
1.2 Effective Kan Fibrations
1.3 Summary of Contents
Part I Π-Types from Moore Paths
2 Preliminaries
2.1 Fibred Structure
2.2 Double Categories of Left and Right Lifting Structures
2.3 Algebraic Weak Factorisation Systems
2.4 A Double Category of Coalgebras
2.5 Cofibrant Generation by a Double Category
2.6 Fibred Structure Revisited
2.7 Concluding Remark on Notation
3 An Algebraic Weak Factorisation System from a Dominance
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
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
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
C Degenerate Horn Fillers Are Unique
D Uniform Kan Fibrations
References
Index
