TITLE: Groupoid structure on G/H?
QUESTION [9 upvotes]: Let $G$ be a group and let $H$ be a subgroup. If $H$ is normal in $G$, then $G/H$ has a group structure. But in general, can there be a groupoid structure on $G/H$(left cosets or right cosets) that generalizes the normal case?

REPLY [2 votes]: Another way of putting this is that there is an equivalences of categories between actions of a group (or groupoid) $G$ and covering morphisms of the group (or groupoid) $G$. See for example 
Higgins, P.J. Notes on categories and groupoids, Mathematical Studies, Volume~32.
Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of
  Categories, No. 7 (2005) pp 1-195.
This links the notion of covering morphisms of groupoids  nicely with the theory of covering spaces.  There one finds that for "nice" spaces $X$ the fundamental groupoid $\pi_1$  gives an equivalence of categories  from covering maps of $X$ to covering morphisms of $\pi_1 X$. Thus a map of spaces is modelled by a morphism of groupoids, and this is convenient. For an exposition in these terms, see my book "Topology and groupoids", or earlier editions (1968, 1988).