Cardinalities in Height 1

TL;DR

This paper develops a framework for understanding cardinalities in stable monoidal p-local infinity categories of height 1, providing explicit formulas for the cardinality of classifying spaces and pi-finite spaces.

Contribution

It introduces a canonical norm map for m-truncated maps, explores properties of cardinality and integration, and derives explicit formulas for cardinalities in specific homotopical contexts.

Findings

Explicit decomposition of cardinality of BG in terms of BC_p

General formula for cardinality of any pi-finite space

Compatibility of cardinality with monoidal structures

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 M\"obius 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 $\pi$-finite space $A$.