  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.
