If the homology of consists of finitely generated projective modules, then the Euler characteristic of is the alternating sum of its Betti numbers (the ranks of its homology modules), if this is finite:
When both of these are defined, they are equal. This is a consequence of the functoriality of the categorical definition (Definition 6).
For a topological space and some ring, its Euler characteristic over is the Euler characteristic, according to Def. 2, of its homology chain complex (for instance singular homology), if this is finite: the alternating sum of its Betti numbers
By default one takes to be the integers, but it is equivalent to take to be the rational numbers. (Choosing a ring with torsion, however, might result in a different “Euler characteristic”.)
This definition is usually known as the Euler-Poincaré formula. Historically earlier was
Let be a finite CW-complex. Write for the set of its -cells. Then the Euler characteristic of is
This is equivalently the Euler characteristic of the cellular chain complex of according to Definition 1. Thus, since the homology of can be computed with cellular homology, this Euler characteristic agrees with the previous one.
Let be a convext polyhedron. Then its Euler characteristic is
All the definitions considered so far can be subsumed by the following general abstract one.
Explicitly, this means the Euler characteristic of an object is the composite
where and are the unit and counit of the dual pair , and is the unit object of the symmetric monoidal category. This subsumes the previous definitions as follows:
In the category of chain complexes over a ring , an object is dualizable if it is finitely generated overall, and projective in each degree. Its dual is then given by . The unit picks out , where encompasses a basis for each and is the dual basis.
The symmetry isomorphism introduces a sign , so that when we then evaluate, we get a contribution of for each of even degree and for each of odd degree. Thus we recover Def. 1. Note that the unit object is itself in degree zero, so that we see only as an element of (so, for instance, the Euler characteristic in this sense of a rank- free -module is zero.
In the derived category of chain complexes over , an object is dualizable if it is quasi-isomorphic to one of the form above. A similar argument shows that its Euler characteristic is then computed as in Def. 2.
The Euler characteristic of a topological space or -groupoid can also be defined directly with this approach, without a detour into homology. Namely, if is any Euclidean Neighborhood Retract?, such as a finite CW complex or a smooth manifold, then its suspension spectrum is dualizable in the stable homotopy category. Its dual is the Thom spectrum of its stable normal bundle, with the unit of the duality being the Thom collapse map. Its categorical Euler characteristic is then an endomorphism of the sphere spectrum, which can be identified with an integer (via ). See around DoldPuppe, corollary 4.6).
Since the categorical definition is purely in terms of the symmetric monoidal structure, Euler characteristics are preserved by any symmetric monoidal functor (as long as enough of its structure maps are isomorphisms). Since chains and homology can be made into symmetric monoidal functors, it follows that all the ways of defining the Euler characteristic of a space agree.
The above Euler characteristic of a topological space is the alternating sum over sizes of homology groups. Similar in construction is the alternating product of sizes of homotopy groups. This goes by the name ∞-groupoid cardinality or homotopy cardinality . But below we shall see that Euler characteristic of higher categories interpolates between this homotopical and the above homological notion.
For a topological space / homotopy type / ∞-groupoid, its homotopy cardinality or ∞-groupoid cardinality is – if it exists – the rational number given by the alternating product of cardinalities of homotopy groups
The process of sending a category to its geometric realization of categories Top Grpd is a way to present topological spaces, and hence ∞-groupoids, by a category: we can think of as the Kan fibrant replacement of : the universal solution to weakly inverting all morphisms of .
Up to the relevant notion of equivalence in an (infinity,1)-category (which is weak homotopy equivalence), every ∞-groupoid arises as the nerve/geometric realization of a category. In fact one can assume the category to be a poset. (This follows from the existence of the Thomason model structure, as discussed in more detail there.)
Since the combinatorial data in a category and all the more in a poset is much smaller than in that of its Kan fibrant replacement , it is of interest to ask if one can read off the Euler characteristic already from itself. This is indeed the case:
If admits both a weighting and a coweighting , then its Euler characteristic is
The definition of Euler characteristic of posets appears for instance in (Rota). For groupoids it has been amplified in BaezDolan. The joint generalization to categories is due to (Leinster), where the above appears as def. 2.2.
Notice that this is in general not an integer, but a rational number. However in sufficiently well-behaved cases, discussed below, coincides with the topological Euler characteristic of its geometric realization. Since that is integral, in these cases also is.
Let be an good enrichment category (for instance a cosmos) which itself comes equipped with a good notion of cardinality, in the form of a monoidal functor
Since strict infinity-categories can be understood as arising from iterative enrichment
etc, this gives a notion of Euler characteristic of strict -categories, hence in particular of strict infinity-groupoids.
One should be able to show that applied to strict -groupoids this does reproduce homotopy cardinality.
Euler characteristic behaves well with respect to the basic operations in homotopy theory.
This is due to (May, 1991).
The following propositions assert that and how the various definitions of Euler characteristics all suitably agree when thez jointly apply.
This appears as (Stanley, 3.8).
The following proposition asserts that the definition 8 of Euler characteristic of a category is indeed consistent, in that it does compute the Euler characteristic, def. 3 of the corresponding -groupoid:
For posets this is due to Philipp Hall, appearing as ([Stanley, prop. 3.8.5]). For finite categories this is (Leinster, cor. 1.5, prop. 2.11).
This is due to … (?)
Mike: Can the Euler characteristic of a category be recovered as the trace for a dualizable object in some symmetric monoidal category?
The latter looks a bit like the “exponential” of the former, so while similar to some extent they are very different notions, taken on face value. Still, the Euler characteristic of a category or rather that of a higher category does interpolate between the two notions:
This is noted in (Leinster, example 2.7).
By prop. 3 we have that the categorical Euler characteristic of is the topological Euler characteristic of . But by prop. 5 we have that the categorical Euler characteristic of is the homotopy cardinality of .
Typically for one and the same -groupoid, Eucler characteristic and homotopy cardinality are never both well defined: if the series for one converges, that for the other does not.
However, by applying some standard apparent “tricks” on non-convergent series, often these can be made sense of after all, and then do agree with the other notion. For more on this see (Baez05).
A standard textbook reference for topological Euler characteristics is page p. 156 and onwards in
Efremovich and Rudyak shown that the Euler characteristic is (up to the overall multiplicative factor) the only additive homotopy invariant of a finite CW complexes:
Behaviour of tracial Euler characteristic under homotopy colimits is discussed in
Textbooks on combinatorial aspects of Euler characteristic include
See Thom spectrum for more on this
The Euler characteristic of groupoids – groupoid cardinality – has been amplified in
An exposition with an eye towards the relation between Euler characteristic and -groupoid cardinality is in
The role of homotopy cardinality in quantization is touched on towards the end of
The generalization of the definition of Euler characteristic from posets to categories is due to
The compatibility of Euler characteristic of categories with homotopy colimits is discussed in
More on Euler characteristics of categories is in
On “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, 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. (arXiv:math.CO/0204009).
James Propp, Euler measure as generalized cardinality . (arXiv).
Euler characteristics of tame spaces:
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 Brown, Euler characteristics, in Cohomology of Groups , Graduate Texts in Mathematics 182, Springer, 1982, pp. 230-272.
O. Y. Viro, Some integral calculus based on Euler characteristic, in Topology and Geometry – Rohlin Seminar, Springer Lec. Notes in Math. 1346, 127–138 (1988) doi
Andreas Blass, Seven trees in one , Jour. Pure Appl. Alg. 103 (1995), 1-21. (web)
Robbie Gates, On the generic solution to in distributive categories , Jour. Pure Appl. Alg. 125 (1998), 191-212.