As explained in that entry, ‘quiver’ is often used to refer to a directed graph, but, as is argued at that entry, this is almost always then used to derive the free category on the that directed graph. The notion of a differential graded quiver, as studied by Lyubashenko and Manzyuk (see reference below), is an enriched analogue of directed graph, enriched over the category of differential graded modules over a commutative ring . (It only uses the underlying category in fact, not the tensor structure). It is introduced precisely to define the analogue of the path category of a directed graph in this enriched context but the structure they define is in fact a linear A-infinity category.
A differential graded -quiver (or simply dg-quiver) is specified by
a small set of objects, denoted ;
for each pair of objects of , a chain complex (= graded -module with differential of degree +1) denoted .
(More work to be done here.)
V. Lyubashenko and O. Manzyuk, Free -categories, Theory and Applications of Categories, Vol. 16, 2006, No. 9, pp 174-205.(pdf)
J. P. May, Operadic categories, -categories, and -categories, Talk given in Morelia, Mexico, May 25, 2001 available at: (pdf)