Gluing pseudo functors via $n$-fold categories

Abstract

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 $F_i:{\mathcal{A}}_i\to \mathcal{D}$ defined on sub-$2$-categories ${\mathcal{A}}_i$ of a $2$-category $\mathcal{C}$, we are concerned with the problem of finding pseudo functors $\mathcal{C}\to \mathcal{D}$ extending $F_i$ 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.

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

References in corpus (3)

• Higher Topos Theory
• Enhanced six operations and base change theorem for higher Artin stacks
• Gluing restricted nerves of $\infty$-categories

Cited by in corpus (3)

• Duality and nearby cycles over general bases
• Gluing restricted nerves of $\infty$-categories
• Segal-type models of higher categories