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groupoid cardinality

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Idea

The homotopy cardinality or \infty-groupoid cardinality of a (sufficiently “finite”) space or ∞-groupoid XX is an invariant of XX (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 XX is the real number

|X|= [x]π 0(X)1|Aut(x)|, |X| = \sum_{[x] \in \pi_0(X)} \frac{1}{|Aut(x)|} \,,

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

If this sum diverges, we say |X|=|X| = \infty. If the sum converges, we say XX 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 XX – equivalently the Euler characteristic of a topological space XX (that’s the same, due to the homotopy hypothesis) – is, if it converges, the alternating product of cardinalities of the (simplicial) homotopy groups

|X|:= [x]π 0(X) k=1 |π k(X,x)| (1) k= [x]1|π 1(X,x)||π 2(X,x)|1|π 3(X,x)||π 4(X,x)|. |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 XX be a discrete groupoid on a finite set SS with nn elements. Then the groupoid cardinality of XX is just the ordinary cardinality of the set SS

    |X|=n. |X| = n \,.
  • Let BG\mathbf{B}G be the delooping of a finite group GG with kk elements. Then

    |BG|=1k |\mathbf{B}G| = \frac{1}{k}
  • More generally, for an action of GG on a set XX, then the cardinality of the action groupoid X//GX//G is |X||G|\frac{\vert X\vert} {\vert G \vert}. This is traditionally sometimes called the class formula.

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

    |B nA|={k if n is even 1k if n is odd |\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)E = core(FinSet) be the groupoid of finite sets and bijections – the core of FinSet. Its groupoid cardinality is the Euler number

    |E|= n1|S n|= n1n!=e. |E| = \sum_{n\in \mathbb{N}} \frac{1}{|S_n|} = \sum_{n\in \mathbb{N}} \frac{1}{n!} = e \,.
  • Let E=(E i)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 vE 0v \in E_0 of EE the cardinality of the set of ii-cells with source vv is independent of the vertex vv. Then the groupoid cardinality of EE can be calculated as |E|= i#(E i) (1) i|E|=\displaystyle{\prod_{i} \#(E_i)^{(-1)^i}}, much like a usual Euler characteristic. For the case when FF is a totally free crossed complex, this gives a very neat formula for the groupoid cardinality of the internal hom HOM(F,E)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 GG a suitable algebraic group, for Σ\Sigma a suitable algebraic curve, and for qq a prime number, then the groupoid cardinality of the 𝔽 q\mathbb{F}_q-points of the moduli stack of G-principal bundles over XX, Bun G(X)Bun_G(X) is the subject of the Weil conjectures on Tamagawa numbers?.

References

Revised on June 24, 2015 15:51:52 by Urs Schreiber (195.113.30.252)