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