  We introduce the category HG, whose objects are topological groupoids endowed
with compatible measure theoretic data: a Haar system and a measure on the unit
space. We then define and study the notion of weak-pullback in the category of
topological groupoids, and subsequently in HG. The category HG is the setting
for topological groupoidification, which we present in separate papers, and in
which the weak pullback is a key ingredient.
