Symmetric effective and degenerate-preferring Kan complexes

Abstract

Kan complexes and fibrations play a fundamental role in simplicial homotopy theory. Recently, the effective Kan fibrations were defined. These are part of a program to reformulate the foundations of simplicial homotopy theory in a constructive setting. For this thesis, we have compared results on Kan fibrations and complexes to the effective Kan fibrations and complexes. We introduce and study two subclasses of the effective Kan fibrations and complexes, namely the symmetric effective and degenerate-preferring Kan fibrations and complexes. We show that simplicial Malcev algebras have the structures of degenerate-preferring Kan complexes.We also show that the symmetric effective Kan fibrations fit into a lifting algebraic weak factorisation system.