Homotopy- and Cohomology Groups of Kan Complexes

TL;DR

This paper introduces new proof techniques for Kan complexes, explores their homotopy and homology groups, constructs Eilenberg-Mac Lane spaces explicitly, and establishes their spectral cohomology equivalence with simplicial cohomology.

Contribution

It provides novel methods for analyzing Kan complexes and explicitly constructs Eilenberg-Mac Lane spaces, linking spectral and simplicial cohomology.

Findings

New proof methods for Kan complexes

Explicit construction of Eilenberg-Mac Lane spaces

Equivalence of spectral and simplicial cohomology

Abstract

This article shows several new methods for proofs on Kan complexes while using them to give a compact introduction to the homotopy groups of these complexes. Then more advanced objects are studied starting with homology and the Hurewicz homomorphism. Eilenberg-Mac Lane-spaces are constructed explicitly and can then be used to define spectral cohomology. In the end the equivalence to the usual simplicial cohomology is shown.