Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

TL;DR

This paper develops algebraic frameworks for two-dimensional holonomy operators using crossed modules of Hopf algebras and associative algebras, introducing exact and blur formulations and their equivalence via fuzzy holonomy.

Contribution

It introduces two formulations of 2D holonomy for Hopf 2-connections and demonstrates their equivalence through fuzzy holonomy, advancing algebraic understanding of higher gauge theories.

Findings

Two formulations of 2D holonomy (exact and blur) are proposed.

The exact and blur holonomies coincide in a natural quotient.

Fuzzy holonomy is defined as the common value of the two formulations.

Abstract

After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of bare algebras.) In particular, we will consider two general formulations of the two-dimensional holonomy of a (fully primitive) Hopf 2-connection (exact and blur), the first being multiplicative the second being additive, proving that they coincide in a certain natural quotient (defining what we called the fuzzy holonomy of a fully primitive Hopf 2-connection).