Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.
There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.
of a group is an interesting construction: it associates to every group 
an abelian group 
(unlike the abelianization ![G/[G, G]](http://s0.wp.com/latex.php?latex=G%2F%5BG%2C+G%5D&bg=ffffff&fg=333333&s=0 "G/[G, G]")
). We might wonder, then, exactly what kind of construction the center is.
we may associate to an object 
its automorphism group 
or endomorphism monoid 
), and this is a canonical construction which again doesn’t extend in an obvious way to a functor 
or 
. (It merely reflects some special part of the 
.) 
, where 
be a finite 
vertices, 
edges, and in general 
different 
-cells. The 
for a convex polyhedron. But where exactly does this expression come from? The modern story involves the 
, but actually one can work on a more intuitive level characterized by the following slogan:
.
, then 
.
whose intersection 
is a subcomplex of both 
and 
. Then 
.
. Part of the reason it is so simple to understand is that 
of its elements, showing that it can be understood either in terms of the group of elements of norm 
in 
(that is, the unitary group 
) or the imaginary subspace of 
-sphere 
. It turns out that there is a picture completely analogous to the picture above, but with 
: that is, 
is isomorphic to the group of elements of norm 
), and there is an exponential map from the imaginary subspace of 
lets us handle elements of 
almost as easily as we handle elements of 
. 
. The group of fractional linear transformations 
acts triple transitively on 
and the stabilizer of any triplet of distinct points is trivial.
preserving the upper half plane 
is 
.
