A groupoid is a category in which every morphism is an isomorphism.

It is easy to see that the category of finite sets with bijections as morphisms forms a groupoid. We will denote this category with .

We can view a set as the groupoid with the sets elements as objects and identity morphism only. Essentially a set is the most boring kind of groupoid we can think about since its points carry no symmetries.

TODO. Essentially just disjoint union.

TODO. Cardinality of coproduct is sum of cardinality.

TODO. Objects are pairs of G and H, morphisms are pairs of morphisms.

TODO. Cardinality of product is product of cardinality.

For the cardinality of a groupoid we pick a representative for each isomorphism class of objects in . Each representative then contributes to the formal sum by , where denotes the cardinality of the automorphism group of x. In symbols

In the case where is finite for all and the series converges we may evaluate it to a real number.

Notice how the definition of groupoid cardinality is crafted explicitly to be an invariant with respect to equivalence of categories.

Proof. TODO

Consider for example the terminal category as well as the groupoid with two isomorphic objects and the obvious 4 morphisms. Both are equivalent as categories and indeed they both have cardinality 1.

Other than sets, proper Groupoids may admit non-integer cardinalities. The groupoid consisting of a single point with 3 automorphisms has cardinality .

The cardinality of the the groupoid of is eulers number .

TODO

TODO