2012/11/30 by Weizhe Zheng
Mathematics · #Adjoint functors #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cohomology #Derived functor #Ext functor #Functor #Functor category #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematics #Natural transformation #Pure mathematics #math.CT #msc:18D05
paper · pdf · doi:10.1007/s40062-016-0126-2
61 pages. This is an updated version of the appendix to arXiv:1006.3810v1. v6: DOI; v5: minor changes; v4: various improvements including change of notation and terminology suggested by the referee; v2: appendix in v1 removed
arxiv created 2016/02/18 · openalex publication_date 2016/03/08 · arxiv updated 2016/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in étale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors Fi\colon Ai→ D defined on sub-2-categories Ai of a 2-category C, we are concerned with the problem of finding pseudo functors C→ D extending Fi up to pseudo natural equivalences. With the help of n-fold categories, we organize gluing data for n pseudo functors into 2-categories. We establish general criteria for equivalence between such 2-categories for n pseudo functors and for n-1 pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in arXiv:1006.3810 to construct extraordinary direct image functors in étale cohomology of Deligne-Mumford stacks.