nLab
groupoid cardinality

Idea

The homotopy cardinality or $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-groupoid. See there for more details.

Definition

Groupoid cardinality

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

$$\mid X\mid =\sum _{[x]\in {\pi }_0(X)}\frac{1}{\mid \mathrm{Aut}(x)\mid },$$|X| = \sum_{[x] \in \pi_0(X)} \frac{1}{|Aut(x)|} \,,

where the sum is over isomorphism classes of objects of $X$ and $\mid \mathrm{Aut}(x)\mid$ is the cardinality of the automorphism group of an object $x$ in $X$.

If this sum diverges, we say $\mid X\mid =\mathrm{\infty }$. If the sum converges, we say $X$ is tame.

$\mathrm{\infty }$-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

$$\mid X\mid :=\sum _{[x]\in {\pi }_0(X)}\prod _{k=1}^{\mathrm{\infty }}\mid {\pi }_k(X,x){\mid }^{(-1{)}^k}=\sum _{[x]}\frac{1}{\mid {\pi }_1(X,x)\mid }\mid {\pi }_2(X,x)\mid \frac{1}{\mid {\pi }_3(X,x)\mid }\mid {\pi }_4(X,x)\mid \cdots .$$|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 total homotopy order of a space, introduced by Quinn in notes in 1995 on TQFTs (see reference list).

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 $S$

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

• Let $BG$ be the delooping of a finite group $G$ with $k$ elements. Then

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

• Let $A$ be an abelian group with $k$ elements. Then we can deloop arbitrarily often and obtain the Eilenberg–Mac Lane objects $B^{n}A$ for all $n\in \mathbb{N}$. (Under the Dold–Kan correspondence $B^{n}A$ 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

  $$\mid B^{n}A\mid =\{\begin{array}{cc}k& \mathrm{if}n\mathrm{is}\mathrm{even}\\ \frac{1}{k}& \mathrm{if}n\mathrm{is}\mathrm{odd}\end{array}$$|\mathbf{B}^n A| = \left\{ \array{ k & if n is \; even \\ \frac{1}{k} & if n is \; odd } \right.

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

  $$\mid E\mid =\sum _{n\in \mathbb{N}}\frac{1}{\mid S_n\mid }=\sum _{n\in \mathbb{N}}\frac{1}{n!}=e.$$|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 $\mid E\mid ={\displaystyle \prod _i\#(E_i{)}^{(-1{)}^i}}$, 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 $\mathrm{HOM}(F,E)$, 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.)

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)

• 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, 1995, Lectures on axiomatic topological quantum field theory , in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory , volume 1 of IAS/Park City Mathematics Series , AMS/IAS.