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The homotopy cardinality or -groupoid cardinality of a (sufficiently “finite”) space or ∞-groupoid is an invariant of (a value assigned to its equivalence class) that generalizes the cardinality of a set (a 0-truncated -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 -groupoid. See there for more details.
The cardinality of a groupoid is the real number
where the sum is over isomorphism classes of objects of and is the cardinality of the automorphism group of an object in .
If this sum diverges, we say . If the sum converges, we say is tame. (See at homotopy type with finite homotopy groups).
This is the special case of a more general definition:
The groupoid cardinality of an ∞-groupoid – equivalently the Euler characteristic of a topological space (that’s the same, due to the homotopy hypothesis) – is, if it converges, the alternating product of cardinalities of the (simplicial) homotopy groups
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.
Let be a discrete groupoid on a finite set with elements. Then the groupoid cardinality of is just the ordinary cardinality of the set
More generally, for an action of on a set , then the cardinality of the action groupoid is . This is traditionally sometimes called the class formula.
Let be an abelian group with elements. Then we can deloop arbitrarily often and obtain the Eilenberg-Mac Lane objects for all . (Under the Dold-Kan correspondence is the chain complex (or depending on notational convention) that is concentrated in degree , where it is the group ). Then
Let be the groupoid of finite sets and bijections – the core of FinSet. Its groupoid cardinality is the Euler number
Let be a finite crossed complex, (i.e., an omega-groupoid; see the work of Brown and Higgins) such that for any object of the cardinality of the set of -cells with source is independent of the vertex . Then the groupoid cardinality of can be calculated as , much like a usual Euler characteristic. For the case when is a totally free crossed complex, this gives a very neat formula for the groupoid cardinality of the internal hom , 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 a suitable algebraic group, for a suitable algebraic curve, and for a prime number, then the groupoid cardinality of the -points of the moduli stack of G-principal bundles over , is the subject of the Weil conjectures on Tamagawa numbers?.
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.
Terence Tao, “Counting objects up to isomorphism: groupoid cardinality” (blog post), 13 April 2017.
Lior Yanovski, Homotopy Cardinality via Extrapolation of Morava-Euler Characteristics (arXiv:2303.02603)
Last revised on March 7, 2023 at 11:43:51. See the history of this page for a list of all contributions to it.