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

We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, 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! The challenge of unifying them remains open.

The Mysteries of Counting - transparencies in PDF format.

• "Negative sets" and Euler characteristic:

• André Joyal, Regle des signes en algebre combinatoire, Comptes Rendus Mathematiques de l'Academie des Sciences, La Societe Royale du Canada VII (1985), 285-290.

• Steve Schanuel, Negative sets have Euler characteristic and dimension, Lecture Notes in Mathematics 1488, Springer Verlag, Berlin, 1991, pp. 379-385.

• Daniel Loeb, Sets with a negative number of elements, Adv. Math. 91 (1992), 64-74.

• Resumming divergent Euler characteristics:

• William J. Floyd and Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1-29.

• R. I. Grigorchuk, Growth functions, rewriting systems and Euler characteristic, Mat. Zametki 58 (1995), 653-668, 798.

• James Propp, Exponentiation and Euler measure, Algebra Universalis 49 (2003), 459-471. Also available as math.CO/0204009.

• James Propp, Euler measure as generalized cardinality. Available as math.CO/0203289.

• Euler characteristics and supersymmetry:

• Matthias Blau and George Thompson, N = 2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant, Comm. Math. Phys. 152 (1993), 41-71.

• Euler characteristics of tame spaces:

• Lou van den Dries, Tame Topology and O-Minimal Structures Cambridge U. Press, Cambridge, 1998. Chapter 4.2: Euler Characteristic.

• Euler characteristics of groups:

• G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.

• Jean-Pierre Serre, Cohomologie des groups discretes, Ann. Math. Studies 70 (1971), 77-169.

• Kenneth S. Brown, Euler characteristics, in Cohomology of Groups, Graduate Texts in Mathematics 182, Springer, 1982, pp. 230-272.

• Euler characteristics of categories and n-categories:

• Tom Leinster, The Euler characteristic of a category. Available as arXiv:math/0610260.

• Tom Leinster, The Euler characteristic of a category as a sum of a divergent series. Available as arXiv:0707.0835.

• Euler characteristics of chain complexes of graded vector spaces:

• Bijective proofs in combinatorics:

• Complex cardinalities:

• Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg. 103 (1995), 1-21. Also available at: http://www.math.lsa.umich.edu/~ablass/cat.html

• Robbie Gates, On the generic solution to P(X) = X in distributive categories, Jour. Pure Appl. Alg. 125 (1998), 191-212.

• Marcelo Fiore and Tom Leinster, An objective representation of the Gaussian integers, Jour. Symb. Comp. 37 (2004), 707-716. Also available as math/0211454.

• Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, Adv. Math. 190 (2005), 264-277. Also available as math/0212377.

• Marcelo Fiore, Isomorphisms of generic recursive polynomial types, to appear in 31st Symposium on Principles of Programming Languages (POPL04). Also available at http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz

• Applications to quantum theory:

• John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. Also available as math/0004133.

• Jeffrey Morton, Categorified algebra and quantum mechanics, TAC 16 (2006), 785-854.
Also try these expository treatments in This Week's Finds in Mathematical Physics:

• Week 147 - Categorification, Euler characteristic versus homotopy cardinality, and the cardinality of the fundamental group of a Riemann surface.

• Week 184 - q-deformed cardinalities: how cardinalities of Grassmannians over finite fields relate to Euler characteristics of Grassmannians over R and C.

• Week 185 - Vector spaces over the field with q elements as a q-deformation of finite sets; more on q-deformed cardinalities.

• Week 186 - The q-polynomial of a simple algebraic group, and using it to compute the cardinalities of flag manifolds over finite fields, and their Euler characteristics over R and C.

• Week 187 - The q-polynomial of a simple algebraic group - the classical groups treated in detail.

• Week 202 - Joyal's species and reasoning with complex cardinalities.

• Week 203 - An object whose cardinality is the golden ratio.

• Week 244 - Leinster's work on the Euler characteristic of a category.
For much more, try these course notes:

Finally, here's an earlier talk on this subject:

The imaginary expression √-a and the negative expression -b resemble each other in that each one, when they seem the solution of a problem, they indicate that there is some inconsistency or nonsense. - Augustus De Morgan

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. - Hermann Weyl