Euler characteristic versus homotopy cardinality

Euler characteristic versus homotopy cardinality

[John Baez]

Dear Categorists -

Some of you might be interested in this talk, since it's secretly
about attempts to categorify the rational numbers.

http://www.math.ucr.edu/home/baez/cardinality/

Euler Characteristic versus Homotopy Cardinality

Abstract:

Just as the Euler characteristic of a space is the alternating sum of the
dimensions of its rational cohomology groups, the homotopy cardinality of
a space is the alternating product of the cardinalities of its homotopy
groups.  The two quantities have many of the same properties, but it's
hard to tell if they're the same, since like Jekyll and Hyde, they're
almost never seen together: there are very few spaces for which the Euler
characteristic and homotopy cardinality are both well-defined. However,
in many cases where one is well-defined, the other may be computed by
dubious manipulations involving divergent series - and the two then agree!
We give examples of this phenomenon and beg the audience to find some
unifying concept which has both Euler characteristic and homotopy cardinality
as special cases.