## Categories in Algebra, Geometry and Mathematical Physics

### Conference in honour of Ross Street's sixtieth birthday

#### July 11-16 2005, Macquarie University, Sydney

Thu, 14 July: 17:00 - 18:00

##### The Mysteries of Counting: Euler Characteristic Versus Homotopy Cardinality

###### Baez, John (University of California, Riverside)

We all know what it means for a set to have 6 elements, but what sort of thing has $-1$ elements, or $5/2$ ?

These questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality $e$, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are 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!

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The challenge of unifying them remains open.

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See \url{http://math.ucr.edu/home/baez/counting/} for talk transparencies and links to further reading material.