Category and Group Rings in Homotopy Theory

Homotopy Coherent Category Theory

Homotopy Theory of Associative Rings

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that similar to spaces the derived category obtained by inverting the quasiisomorphisms is natura...

Homotopy Theory and TDA with a View Towards Category Theory

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view. In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories wit...

Symmetric Ideals in Group Rings and Simplicial Homotopy

In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described with the help of homotopy groups of spheres.

Category O over Skew Group Rings

We study the BGG Category O over a skew group ring, involving a finite group acting on a regular triangular algebra. We relate the representation theory of the algebra to Clifford theory for the skew group ring, and obtain results on block decomposition, semisimplicity, and enough projectives. O is also shown to be a highest weight category; the BGG Reciprocity formula is slightly different bec...