Cardinalities in Height 1
Yifan Li
TL;DR
The paper develops a framework for ambidexterity and higher semiadditivity in stable $p$-local $\,\infty$-categories by constructing canonical norm maps for $m$-finite spaces and defining a robust integration theory. It then leverages Möbius inversion and the Burnside ring to express cardinalities of classifying spaces and more general $\,\pi$-finite spaces, with a central focus on height-1 (i.e., $\,|BG|$) cases. The main contributions include inductive construction of canonical norms, a detailed height/mode theory, and explicit, computable formulas for $|BG|$, $|BC_p|$, and general $\,|A|$ in terms of subgroup data and Postnikov decompositions. These results bridge ambidexterity with concrete numerical invariants in $p$-local settings, enabling a partition-of-unity style decomposition of cardinalities that parallels classical group-theoretic counting. The work advances a calculational toolkit for semiadditive height and paves the way for broader applicability to $\,\pi$-finite spaces and chromatic-like phenomena in homotopy theory.
Abstract
In this article, we give an introduction to the notion of ambidexterity and norm map, and construct inductively the canonical norm map for $m$-truncated maps for some $m\geq-1$, on which the definitions of integration and cardinality are built. We then use several propositions to justify the properties of cardinality and integration and their compatibility with monoidal structure. We give a brief introduction of the definition and behaviors of semiadditive height. Focusing on stable monoidal $p$-local $\infty$-categories of height 1, for any finite group $G$, with the help of Möbius function and Burnside ring, we give an explicit decomposition of the cardinality of $BG$ into an expression of the cardinality of $BC_p$. Eventually, we generalize the result and conclude with a formula of the cardinality of any $π$-finite space $A$.