Cardinalities in Height 1

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 $m\ge -1$, 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 $B{C}_p$. Eventually, we generalize the result and conclude with a formula of the cardinality of any $\ddot{I}€$-finite space $A$.

45 pages