homotopy theory, (∞,1)-category theory, homotopy type theory
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The homotopy cardinality or $\infty$-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 $\infty$-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.
The cardinality of a groupoid $X$ is the real number
where the sum is over isomorphism classes of objects of $X$ and $|Aut(x)|$ is the cardinality of the automorphism group of an object $x$ in $X$.
If this sum diverges, we say $|X| = \infty$. If the sum converges, we say $X$ 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 $X$ – 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
This corresponds to what is referred to as the homotopy Euler characteristic by Kontsevich (1988) and the total homotopy order of a space by Quinn (1995), although similar ideas were explored by several researchers at that time.
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 $S$
Let $\mathbf{B}G$ be the delooping of a finite group $G$ with $k$ elements. Then
More generally, for an action of $G$ on a set $X$, then the cardinality of the action groupoid $X//G$ is $\frac{\vert X\vert} {\vert G \vert}$. This is traditionally sometimes called the class formula.
Let $A$ be an abelian group with $k$ elements. Then we can deloop arbitrarily often and obtain the Eilenberg-Mac Lane objects $\mathbf{B}^n A$ for all $n \in \mathbb{N}$. (Under the Dold-Kan correspondence $\mathbf{B}^n A$ 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
Let $E = core(FinSet)$ be the groupoid of finite sets and bijections – the core of FinSet. Its groupoid cardinality is the Euler number
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 $|E|=\displaystyle{\prod_{i} \#(E_i)^{(-1)^i}}$, 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 $HOM(F,E)$, 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 $G$ a suitable algebraic group, for $\Sigma$ a suitable algebraic curve, and for $q$ a prime number, then the groupoid cardinality of the $\mathbb{F}_q$-points of the moduli stack of G-principal bundles over $X$, $Bun_G(X)$ 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)
Maxim Kontsevich, Rational conformal field theory and invariants of 3-dimensional manifolds, Centre de Physique Théorique Marseille, preprint 2189, (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, Lectures on axiomatic topological quantum field theory, in Dan Freed, Karen Uhlenbeck (eds.) Geometry and Quantum Field Theory 1 (1995) [doi:10.1090/pcms/001]
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 September 25, 2024 at 13:28:37. See the history of this page for a list of all contributions to it.