Ge-categories and applications Ge-categories, i.e., categores enriched over groupoids (these are 2-categories where the set of morhisms $HOM(a,b)$ has a groupoid structure) seem to be useful in homotopy theory. 
Question: What are results in homotopy theory proved using ge-categories?
I am especially interested in ge-categories where 1-morphisms are not invertible. I would also appreciate references. 
 A: There are lots of results provable in this context. In the book I wrote with Heiner Kamps (which is easily found via Google so I won't advertise here!) we looked at the problem of what results in homotopy theory could be proved with a restricted set of fillers for boxes in a cubical enrichment of a category.  This applies to your question since groupoid enriched categories give rise to such cubical homotopy theories very easily.  
There is an old paper: P. H. H. Fantham and E. J. Moore, Groupoid enriched categories and homotopy theory, Canad. J. Math., 35, (1983), 385 – 416, which also examines this question and of course, some of the classical book by Gabriel and Zisman is devoted to developing GE-categories in your sense.
As Noah points out, these 2-categories are nowadays more often called (strict) (2,1)-categories although that term (without the `strict') also is used for bicategories in which the homs are groupoids. Try looking up locally groupoidal 2-category in the nLab for more on that side of things.
(Edited (08-01-2018): I should have mentioned the extensive work by Hans Baues and his coworkers on what he calls 'track categories'.  These are the 'ge-categories' of the question. There are many problems solved within the more calculative part of homotopy theory that are stated in terms of these track categories but which have direct interpretation in more classical approaches homotopy theory.)
A: I claim that a good example of the use of Grpd-enriched categories is the Murayama-Shimakawa model for equivariant classifying spaces for equivariant principal bundles:
Such equivariant classifying spaces were abstractly characterized by the original authors (tom Dieck, Bierstone, Lashof, May) but remained somewhat impalpable until Murayama & Shimakawa 1995 presented a concrete model based on topological realization of equivariant topological mapping groupoids. This construction was only more recently highlighted by Guillou, May & Merling 2017 for its neat category-theoretic nature.
While none of these authors makes the ambient Grpd-enriched category theory fully explicit, one recognizes it behind the scenes.
(For instance, the groupoid of "crossed homomorphism" is equivalently simply the mapping groupoid of sections of the projection out of the delooping groupoid of the corresponding semidirect product group...)
We are writing up a systematic account of universal equivariant principal bundle theory as an excercise in Grpd-enriched category theory. Will add a pointer when its ready for public consumption.
