I'm not sure the literature is entirely consistent on the use of these terms.
Here are some ideas I learned from Jean Pradines' explanations in 1981 in Toulouse of his note
- Pradines, J., Th\'eorie de Lie pour les groupoides diff\'erentiables, relation entre propri\'et\'es locales et globales, Comptes Rendus Acad. Sci. Paris, S\'er A, 263 (1966), 907-910.
the first of his 4 notes introducing the relation between Lie groupoids and Lie algebroids. The ideas for the first note were written up in detail in various work by research students at Bangor, with the full knowledge of Pradines. (Kirill Mackenzie worked quite independently on the succeeding theory of Lie groupoids and Lie algebroids, published in his 1987 book.)
Intuitively, non trivial holonomy may be explained as an iteration of local procedures which return to the starting point with a change of phase. This idea is related to hysteresis, and shows the nice relation with physics. The problem is to define rigorously all the terms used in this explanation!
A Monodromy Principle is enunciated in Chevalley's famous book ``Lie groups"; the Principle may be explained as giving an extension of a restriction of a local morphism to a morphism on a simply connected cover.
In foliation theory, it is usual to define a monodromy groupoid as the disjoint union of the fundamental groupoids of the leaves, with a topology reflecting the local structure of the foliation; and to define the holonomy groupoid as a quotient of the monodromy groupoid. However this does not easily yield a Monodromy Principle.
Pradines' idea for his Th\'eor`eme 2 in the Note was to use the Monodromy Principle as guiding the construction of a Monodromy Groupoid of a Lie groupoid, generalising the universal cover of a connected Lie group. So, given a neighbourhood $W$ of the identities of a Lie groupoid, one forms the groupoid $M(W)$ which is universal for all local morphisms of $W$ into groupoids. The problem is to define an appropriate topology on $M(W)$ and Pradines solves this using a notion of holonomy groupoid, although in this case the holonomy is trivial (!).
In the case of a Lie group $G$, the topology on $G$ is defined by a neighbourhood of the identity satisfying some reasonable conditions given in, for example, Bourbaki. Now one can define a local Lie groupoid to be a groupoid $G$ with a set $W$ containing the identities and satisfying a number of reasonable conditions. However it is no longer true that the topology of $W$ extends to a topology on $G$ making it a Lie groupoid. Instead there is, under reasonable conditions, a Holonomy Groupoid $Hol(G,W)$ which projects to $G$ and which has a Lie groupoid structure locally like $W$. The construction of Pradines is written up in:
- Aof, M.E.-S.A.-F. and Brown, R., The holonomy groupoid of a locally topological groupoid, Top. Appl. 47, 1992, 97-113
(with the agreement of Pradines). It really does use the idea of ``iteration of local procedures" where the local procedures here are given by Ehresmann's local admissible sections of $G$ with values in $W$. The holonomy groupoid $Hol(G,W)$ has a universal property for maps of Lie groupoids into $G$.
The application to the monodromy groupoid is written up in: Brown, R. and Mucuk, O., The monodromy groupoid of a Lie groupoid, Cah. Top. G\'eom. Diff. Cat. 36 (1995) 345-369.
See also: Mucuk, O., Kılı¸carslan, B., S¸ahan, T. and Alemdar N. Group-groupoid and monodromy groupoid, Topology and its Applications 158 (2011) 2034-2042.
So holonomy comes out as a kind of right adjoint, and monodromy as a kind of left adjoint, which explains one difference. But there seems still work to do to explain everything stated in the two Theorems of Pradines' first Note, and to apply these ideas more widely. This is a reason for adverting advertising Pradines' ideas here.
