On a characterization of Higher Semiadditivity
Areeb Shah Mohammed
Universität Regensburg, Nov 2022
Survey article: all results are generally already known. Contains a few alternate proofs in an attempt to work as much as possible with quasicategories as opposed to simplicial categories or complete Segal spaces.
Michael Hopkins and Jacob Lurie introduce for m at most n, a notion of m-semiadditivity. This generalizes the classical notion of a semiadditive (infinity) category. Intuitively, m-semiadditive infinity categories are those in which limits and colimits of diagrams indexed by m-finite spaces (that is, m-finite infinity groupoids) are canonically equivalent. Yonatan Harpaz proves a universal property of the infinity category of spans of n-finite spaces with m-truncated wrong way maps. This is used to establish an equivalent characterization of m-semiadditivity in terms of a well behaved, essentially unique action of this category of spans. This has the advantage of not only providing a more succinct method of detecting m-semiadditivity, but also providing a versatile structure to work with m-semiadditive infinity categories. In this thesis, we survey this sequence of results.