The cardinality of a metric space


Venue   87th Peripatetic Seminar on Sheaves and Logic, Patras, 23/3/08

Abstract   In the last couple of years it has become apparent that there is a sensible definition of the "Euler characteristic" or "cardinality" of a category. This notion extends easily to enriched categories, and in particular to metric spaces. I will explain what the cardinality of a metric space is, and describe some relations with convex geometry and geometric measure theory.

Slides   In this pdf file (1MB).

An earlier and very productive discussion of these ideas can be found at the n-Category Café.

Errata   The scan on page 11 went a bit wrong. The second half of the first line should read "Then |x + y| = |x|.|y|.", and the first line of the definition should read "Let A = {a1, …, an} be a finite metric space."

In the example on page 11, the cardinality of A should be 2/(1 + e-2d).

In the graph on page 12, the vertical axis should be labelled |tA|.

In the first conjecture on page 16, "connected" should be "convex" (in a suitable sense). The first picture is inappropriate.

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