nLab groupoid cardinality

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

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 by Quinn (1995), although similar ideas were explored by several researchers at that time.

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

Last revised on March 29, 2023 at 12:46:52. See the history of this page for a list of all contributions to it.