This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

Higher Topos Theory

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The Geometry and Cohomology of Some Simple Shimura Varieties.

In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular. We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ1(N) denote the subgroup of SL2(Z) consisting of matrices that modulo N are of the form ( 1 ∗ 0 1 ) .

https://dash.harvard.edu/bitstream/handle/1/3626807/Taylor_Wild3AdicExercises.pdf;sequence=2

On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

The study of topological quantum field theories increasingly relies upon concepts from higher‐dimensional algebra such as n‐categories and n‐vector spaces. We review progress towards a definition of n‐category suited for this purpose, and outline a program in which n‐dimensional topological quantum field theories (TQFTs) are to be described as n‐category representations. First we describe a ‘‘suspension’’ operation on n‐categories, and hypothesize that the k‐fold suspension of a weak n‐category stabilizes for k≥n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n‐dimensional unitary extended TQFTs as weak n‐functors from the ‘‘free stable weak n‐category with duals on one object’’ to the n‐category of ‘‘n‐Hilbert spaces.’’ We conclude by describing n‐categorical generalizations of deformation quantization and the quantum double construction.

Higher dimensional algebra and topological quantum field theory

Théorie de Dieudonné cristalline II

On the classification of 2-gerbes and 2-stacks

We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J.- L. Brylinski and N. Hitchin. We give a global definition of the 3-curvature of this connective structure as a 3-form on X with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebra-valued 3-forms. The Bianchi identity, which the curvature of a connection on a principal bundle satisfies, is replaced here by a more elaborate equation.

Differential Geometry of Gerbes

Differential geometry of gerbes

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1992, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Lawrence Breen

Papers

Théorie de Dieudonné cristalline II

On the classification of 2-gerbes and 2-stacks

Differential Geometry of Gerbes

Differential geometry of gerbes

Théorie de Schreier supérieure