2021/08/28 by Axel Osmond, Osmond, Axel
Mathematics · #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2108.12697
openalex publication_date 2021/08/28 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor, categorifying an analog 1-categorical result. We also describe further laxness conditions, giving some properties of the already known lax familial pseudofunctors. We also describe 2-dimensional analogs of orthogonality and factorization systems, and use them to construct examples of bistable pseudofunctors through inclusion of left objects and left maps. We apply the latter construction to several examples of factorization systems for geometric morphisms to produce geometry-like situations for Grothendieck topoi. In particular we prove a new (terminally connected, pro-etale) factorization theorem for geometric morphisms, which corresponds to a certain local geometry involved in the general construction of spectra.