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

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).