Gluing pseudo functors via nn-fold categories

Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in \'etale 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 $F_i\colon \mathcal{A}_i\to \mathcal{D}$ 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 \'etale cohomology of Deligne-Mumford stacks.Comment: 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 remove

Companions on Artin stacks

Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's theorem to smooth Artin stacks and deduce Deligne's conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.Comment: 26 pages. v7: fixed typos, to appear in Math.

Enhanced adic formalism and perverse t-structures for higher Artin stacks

In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic formalism for \'etale cohomology of Artin stacks and prove several desired properties including the base change theorem. In addition, we define perverse t-structures on Artin stacks for general perversity, extending Gabber's work on schemes. Our results generalize results of Laszlo and Olsson on adic formalism and middle perversity. We continue to work in the world of $\infty$-categories in the sense of Lurie, by enhancing all the derived categories, functors, and natural transformations to the level of $\infty$-categories.Comment: 53 pages. v2: reformulatio

Categorical traces and a relative Lefschetz-Verdier formula

We prove a relative Lefschetz-Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$-category of cohomological correspondences. We show that local acyclicity is equivalent to dualizability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalizing a theorem of Gabber.Comment: 26 pages. v3: minor improvement