  The existence of interesting multiplicative cohomology theories for orbifolds
was first suggested by string theorists. Orbifold products have been intensely
studied by mathematicians for the last fifteen years. In this paper we focus on
the virtual orbifold product that was first introduced in Lupercio et al.
(2007). We construct a categorification of the virtual orbifold product that
leverages the geometry of derived loop stacks. By work of Ben-Zvi, Francis and
Nadler, this reveals connections between virtual orbifold products and Drinfeld
centers of monoidal categories, thus answering a question of Hinich.
