  It is well-known that for certain local connectivity assumptions the
fundamental groupoid of a topological space can be equipped with a topology
making it a topological groupoid. In other words, the fundamental groupoid
functor can be lifted through the forgetful functor from topological groupoids
to groupoids. This article shows that for a map $Y \to X$ with certain relative
local connectivity assumptions, the fibrewise fundamental groupoid can also be
lifted to a topological groupoid over the space $X$. This allows the
construction of a simply-connected covering space in the setting of fibrewise
topology, assuming a local analogue of the definition of an ex-space. When
applied to maps which are up-to-homotopy locally trivial fibrations the result
is a categorified version of a covering space. The fibrewise fundamental
groupoid can also be used to define a topological fundamental bigroupoid of a
(suitably locally connected) topological space.
