  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).
