Following the logic of space and quantity this may be understood as saying that a bare ∞-groupoid without further structure gives just a prescription for how to map the point into it: there is an -groupoid of ways of mapping the point into the -groupoid , and that reproduces .
A Lie ∞-groupoid – or ∞-Lie groupoid as we shall say – should instead be an -groupoid that comes with the additional information on what a (contractible) smooth family of points inside it should be. Accordingly, it should provide a rule that assigns to each (contractible) smooth family of points an -groupoid of smooth maps of into .
Notice that an -groupoid that may be probed by contractible ordinary manifolds is slightly more general than being an -groupoid internal to diffeological spaces. Therefore what we call -Lie groupoids here are considerably more general than some notion of groupoids internal to manifolds. We shall still just say -Lie groupoid for our definition, for brevity.
Our category of -Lie groupoids is a nice category that contains some pathological objects:
supports a good general ∞-Lie theory
while restriction to special nice objects is a matter of concrete applications.
The restriction that be a site of smooth loci is to ensure that there is a good notion of infinitesimal path ∞-groupoid? in . But all the common Models for Smooth Infinitesimal Analysis are of this form.
and hence to an ∞-functor
on the (∞,1)-topos.