Journal of Homotopy and Related Structures

, Volume 12, Issue 1, pp 189–271 | Cite as

Gluing pseudo functors via n-fold categories

  • Weizhe ZhengEmail author


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 \(F_i:\mathcal {A}_i\rightarrow \mathcal {D}\) defined on sub-2-categories \(\mathcal {A}_i\) of a 2-category \(\mathcal {C}\), we are concerned with the problem of finding pseudo functors \(\mathcal {C}\rightarrow \mathcal {D}\) 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 Zheng (Sci China Math 58(3):565–632, 2015) to construct extraordinary direct image functors in étale cohomology of Deligne-Mumford stacks.


Pseudo functor Gluing data 2-Category n-Fold category Hypercube 

Mathematics Subject Classification




In the course of this work the author has benefited very much from conversions with Yifeng Liu. The author also thanks Joseph Ayoub, Johan de Jong, Ofer Gabber, and Luc Illusie for useful discussions. The author is grateful to the referees for the numerous comments they made on previous versions of this article. This work was partially supported by China’s Recruitment Program of Global Experts; National Natural Science Foundation of China Grant 11321101; Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences; National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.


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Copyright information

© Tbilisi Centre for Mathematical Sciences 2016

Authors and Affiliations

  1. 1.Morningside Center of MathematicsAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

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