Cardinalities in Height 1

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≥-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 ∞-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 BCₚ. Eventually, we generalize the result and conclude with a formula of the cardinality of any π-finite space A.