Effective Kan Fibrations in Simplicial Sets

1. Front Matter

   Pages i-x

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2. Introduction

   • Benno van den Berg, Eric Faber

   Pages 1-9

3. Π-Types from Moore Paths

   1. Front Matter

      Pages 11-11

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   2. Preliminaries

      • Benno van den Berg, Eric Faber

      Pages 13-41

   3. An Algebraic Weak Factorisation System from a Dominance

      • Benno van den Berg, Eric Faber

      Pages 43-48

   4. An Algebraic Weak Factorisation System from a Moore Structure

      • Benno van den Berg, Eric Faber

      Pages 49-73

   5. The Frobenius Construction

      • Benno van den Berg, Eric Faber

      Pages 75-86

   6. Mould Squares and Effective Fibrations

      • Benno van den Berg, Eric Faber

      Pages 87-97

   7. Π-Types

      • Benno van den Berg, Eric Faber

      Pages 99-103

4. Simplicial Sets

   1. Front Matter

      Pages 99-99

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   2. Effective Trivial Kan Fibrations in Simplicial Sets

      • Benno van den Berg, Eric Faber

      Pages 107-118

   3. Simplicial Sets as a Symmetric Moore Category

      • Benno van den Berg, Eric Faber

      Pages 119-135

   4. Hyperdeformation Retracts in Simplicial Sets

      • Benno van den Berg, Eric Faber

      Pages 137-165

   5. Mould Squares in Simplicial Sets

      • Benno van den Berg, Eric Faber

      Pages 167-179

   6. Horn Squares

      • Benno van den Berg, Eric Faber

      Pages 181-203

   7. Conclusion

      • Benno van den Berg, Eric Faber

      Pages 205-206

5. Back Matter

   Pages 207-232

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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.

• Simplicial Sets
• Homotopy Theory
• Constructive Mathematics
• Homotopy Type Theory
• Kan Complexes

• Institute for Logic, Language and Computation, Universiteit van Amsterdam, Amsterdam, The Netherlands

  Benno van den Berg

• Frankfurt am Main, Germany

  Eric Faber

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