nLab
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, which occurs notably in notes Frank Quinn in 1995 on TQFTs (see reference list), although similar ideas were explored by several researchers at that time.

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References