2-categories and cyclic homology

TL;DR

This thesis explores how distributive laws between comonads can be used to understand and extend cyclic homology theories for associative and Hopf algebras, using 2-categorical methods.

Contribution

It introduces a 2-categorical framework for cyclic homology, generalizes twisting procedures, and characterizes categories with duplicial structures.

Findings

Cyclic homology arises from distributive laws between comonads.

Extension of twisting cyclic homology to duplicial objects via distributive laws.

Characterization of categories whose nerve admits a duplicial structure.

Abstract

The topic of this thesis is the application of distributive laws between comonads to the theory of cyclic homology. Explicitly, our main aims are: 1) To study how the cyclic homology of associative algebras and of Hopf algebras in the original sense of Connes and Moscovici arises from a distributive law, and to clarify the role of different notions of bimonad in this generalisation. 2) To extend the procedure of twisting the cyclic homology of a unital associative algebra to any duplicial object defined by a distributive law. 3) To study the universality of Bohm and Stefan's approach to constructing duplicial objects, which we do in terms of a 2-categorical generalisation of Hochschild (co)homology. 4) To characterise those categories whose nerve admits a duplicial structure.