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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.
List of contents
- 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.
