nLab groupoid cardinality

Idea

The homotopy cardinality or $\infty$-groupoid cardinality of a (sufficiently “finite”) space or ∞-groupoid $X$ is an invariant of $X$ (a value assigned to its equivalence class) that generalizes the cardinality of a set (a 0-truncated $\infty$-groupoid).

Specifically, whereas cardinality counts elements in a set, the homotopy cardinality counts objects up to equivalences, up to 2-equivalences, up to 3-equivalence, and so on.

This is closely related to the notion of Euler characteristic of a space or $\infty$-groupoid. See there for more details.

Definition

Groupoid cardinality

The cardinality of a groupoid $X$ is the real number

$$|X| = \sum_{[x] \in \pi_0(X)} \frac{1}{|Aut(x)|} \,,$$

where the sum is over isomorphism classes of objects of $X$ and $|Aut(x)|$ is the cardinality of the automorphism group of an object $x$ in $X$.

If this sum diverges, we say $|X| = \infty$. If the sum converges, we say $X$ is tame. (See at homotopy type with finite homotopy groups).

$\infty$-Groupoid cardinality

This is the special case of a more general definition:

The groupoid cardinality of an ∞-groupoid $X$ – equivalently the Euler characteristic of a topological space $X$ (that’s the same, due to the homotopy hypothesis) – is, if it converges, the alternating product of cardinalities of the (simplicial) homotopy groups

$$|X| := \sum_{[x] \in \pi_0(X)}\prod_{k = 1}^\infty |\pi_k(X,x)|^{(-1)^k} = \sum_{[x]} \frac{1}{|\pi_1(X,x)|} |\pi_2(X,x)| \frac{1}{|\pi_3(X,x)|} |\pi_4(X,x)| \cdots \,.$$

This corresponds to what is referred to as the homotopy Euler characteristic by Kontsevich (1988) and the total homotopy order of a space by Quinn (1995), although similar ideas were explored by several researchers at that time.

Examples

• Let $X$ be a discrete groupoid on a finite set $S$ with $n$ elements. Then the groupoid cardinality of $X$ is just the ordinary cardinality of the set $S$

  $$|X| = n \,.$$

• Let $\mathbf{B}G$ be the delooping of a finite group $G$ with $k$ elements. Then

  $$|\mathbf{B}G| = \frac{1}{k}$$

• More generally, for an action of $G$ on a set $X$, then the cardinality of the action groupoid $X//G$ is $\frac{\vert X\vert} {\vert G \vert}$. This is traditionally sometimes called the class formula.

• Let $A$ be an abelian group with $k$ elements. Then we can deloop arbitrarily often and obtain the Eilenberg-Mac Lane objects $\mathbf{B}^n A$ for all $n \in \mathbb{N}$. (Under the Dold-Kan correspondence $\mathbf{B}^n A$ is the chain complex $A[n]$ (or $A[-n]$ depending on notational convention) that is concentrated in degree $n$, where it is the group $A$). Then

  $$|\mathbf{B}^n A| = \begin{cases} k & \text{if }\; n \;\text{ is even} \\ \frac{1}{k} & \text{if }\; n \;\text{ is odd} \end{cases}$$

• Let $E = core(FinSet)$ be the groupoid of finite sets and bijections – the core of FinSet. Its groupoid cardinality is the Euler number

  $$|E| = \sum_{n\in \mathbb{N}} \frac{1}{|S_n|} = \sum_{n\in \mathbb{N}} \frac{1}{n!} = e \,.$$

• Let $E=(E_i)$ be a finite crossed complex, (i.e., an omega-groupoid; see the work of Brown and Higgins) such that for any object $v \in E_0$ of $E$ the cardinality of the set of $i$-cells with source $v$ is independent of the vertex $v$. Then the groupoid cardinality of $E$ can be calculated as $|E|=\displaystyle{\prod_{i} \#(E_i)^{(-1)^i}}$, much like a usual Euler characteristic. For the case when $F$ is a totally free crossed complex, this gives a very neat formula for the groupoid cardinality of the internal hom $HOM(F,E)$, in the category of omega-groupoids. Therefore the groupoid cardinality of the function spaces (represented themselves by internal homs) can easily be dealt with if the underlying target space is represented by a omega-groupoid, i.e., has trivial Whitehead products. (This is explored in the papers by Faria Martins and Porter mentioned in the reference list, below.)

• for $G$ a suitable algebraic group, for $\Sigma$ a suitable algebraic curve, and for $q$ a prime number, then the groupoid cardinality of the $\mathbb{F}_q$-points of the moduli stack of G-principal bundles over $X$, $Bun_G(X)$ is the subject of the Weil conjectures on Tamagawa numbers.

Related concepts

• class formula

References

• John Baez, Alexander Hoffnung, Christopher Walker, Groupoidification Made Easy (web pdf, blog); Higher-dimensional algebra VII: Groupoidification, arxiv/0908.4305

• John Baez, James Dolan, From Finite Sets to Feynman Diagrams (arXiv)

• João Faria Martins, On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex (web pdf )

• João Faria Martins, Tim Porter, On Yetter’s Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups, (web pdf)

• Maxim Kontsevich, Rational conformal field theory and invariants of 3-dimensional manifolds, Centre de Physique Théorique Marseille, preprint 2189, (web pdf)

• Tom Leinster, The Euler characteristic of a category (arXiv, TWF, blog, blog)

• Kazunori Noguchi, The Euler characteristic of infinite acyclic categories with filtrations, arxiv/1004.2547

• Frank Quinn, Lectures on axiomatic topological quantum field theory, in Dan Freed, Karen Uhlenbeck (eds.) Geometry and Quantum Field Theory 1 (1995) [doi:10.1090/pcms/001]

• Terence Tao, “Counting objects up to isomorphism: groupoid cardinality” (blog post), 13 April 2017.

• Lior Yanovski, Homotopy Cardinality via Extrapolation of Morava-Euler Characteristics. Selecta Mathematica, Volume 29, article number 81, (2023) (arXiv:2303.02603)