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.
Attempts to unify Euler characteristic and homotopy cardinality:
John D. Berman, Euler
characteristic and homotopy cardinality. (Apparently the negative
result in this paper is correct but the proof of the positive result
has an error, and the result may not even be true: there's a footnote in
the next paper concerning this.)
André Joyal, Regle des signes en algebre combinatoire, Comptes
Rendus Mathematiques de l'Academie des Sciences, La Societe Royale
du CanadaVII (1985), 285-290.
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. Zametki58 (1995), 653-668, 798.
James Propp, Exponentiation and Euler measure, Algebra Universalis49 (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. Studies70 (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:
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.
Generalized sets with nonnegative real cardinalities:
George Janelidze and Ross Street, Real sets,
Tbilisi Mathematical Journal10 (2017), 23-49.
Corrected version available as arXiv:1704.08787.
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.
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:
John Baez and Derek Wise,
Quantization and Categorification.
John Baez, Euler characteristic versus homotopy cardinality,
lecture at the Fields Institute Program on Applied Homotopy Theory,
September 20, 2003. Available in PDF form
at: http://www.math.ucr.edu/home/baez/cardinality/
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
