The cardinality of a set is just the number of elements. 


To make sense of the cardinality of a category, one has to account for the morphisms. The usual definition is the sum over the isomorphism classes of 1/#automorphisms.



> 
> What's the cardinality of a higher category?
> 
> 
> 


I.e., should I somehow take into account the automorphisms between the automorphisms, etcetera?



What's the cardinality of a higher category?
Note that the formula "sum 1/#automorphisms" only depends on the (maximal sub-)groupoid of the category.


So your question becomes: what's a good notion of cardinality for $n$-groupoids (a.k.a. spaces with vanishing $\pi\_k$ for all $k>n$)?


One possible answer is to take the sum over connected components of the alternating product of the cardinality of the homotopy groups:


$$
\sum\_{components} \frac1{|\pi\_1|}\cdot |\pi\_2|\cdot \frac1{|\pi\_3|}\cdot |\pi\_4|\cdot\ldots
$$


