Higher Topos Theory

Higher Topos Theory
Name	Higher Topos Theory
Author	Jacob Lurie
Country	United States
Language	English
Subject	Mathematics
Publisher	Princeton University Press
Pub date	2009
Pages	520
Isbn	9780691140490

Higher Topos Theory. Higher Topos Theory is a foundational work linking modern algebraic topology with category theory through the development of ∞-categorical techniques, written by Jacob Lurie and published by Princeton University Press. The book systematically develops the theory of ∞-categories, model structures, and Grothendieck ∞-topoi, connecting to programs pursued in the contexts of the Institute for Advanced Study, the Clay Mathematics Institute, and contemporary research influenced by figures such as Grothendieck, Quillen, Boardman, Vogt, and Boardman–Vogt-style homotopy theory. It has influenced work at institutions like Harvard University, Massachusetts Institute of Technology, Princeton University, University of Chicago, and laboratories including Microsoft Research.

Introduction

Higher Topos Theory recasts classical notions from Alexandre Grothendieck's theory of topoi and Grothendieck school into the language of ∞-categories, synthesizing ideas from Daniel Quillen's model categories, Michael Boardman's homotopy theoretic frameworks, and the homotopical algebra developed by Jean-Pierre Serre, Henri Cartan, and Samuel Eilenberg. Lurie formalizes ∞-categorical limits and colimits, mapping spaces, and Cartesian fibrations influenced by constructions in the work of Grothendieck, Pierre Deligne, Jean-Louis Verdier, and André Joyal. The text has deep ties to seminars and conferences at venues such as the Simons Center for Geometry and Physics, Banff International Research Station, and the Institute for Advanced Study.

Foundations: ∞-Categories and Model Structures

The foundations lay out the theory of quasi-categories (also called weak Kan complexes) building on contributions by André Joyal, connecting to model structures introduced by Daniel Quillen and compared with the homotopy coherent nerve of Cordier and Porter and the nerve functors used in work by Boardman and Vogt. Lurie develops comparisons with the Bergner model structure, the Rezk model, and the work of Charles Rezk and Julie Bergner, situating quasi-categories amid simplicial sets and simplicial categories studied at École Normale Supérieure and in seminars influenced by Jean-Pierre Serre and Alexandre Grothendieck. The exposition uses tools related to the Dold–Kan correspondence and references techniques reminiscent of Serre spectral sequence computations and comparisons to constructions by Pierre Deligne.

Higher Sheaves and Grothendieck ∞-Topoi

Lurie defines Grothendieck ∞-topoi as left exact reflective localizations of presheaf ∞-categories, building on the philosophy of Alexandre Grothendieck’s original topoi and extending ideas from Artin and Verdier in the context of étale cohomology and stacks developed by Deligne and Giraud. The treatment connects to stacks and descent theory shaped by lectures at institutions such as École Normale Supérieure and the Institut des Hautes Études Scientifiques where figures like Jean Giraud and Pierre Deligne influenced modern formulations. The chapter elaborates analogues of Giraud’s axioms and relates to the theory of topoi appearing in work at Princeton University and collaborations with researchers affiliated to Harvard University and University of Chicago.

Geometric Morphisms and Descent

Lurie generalizes geometric morphisms between ∞-topoi, formulating left and right adjoint functor pairs and higher versions of Grothendieck’s descent conditions, tracing intellectual lineage to Grothendieck’s SGA seminars and descent ideas used by Pierre Deligne and Jean-Louis Verdier. This development aligns with derived algebraic geometry programs advanced at the Institute for Advanced Study, Harvard University, and by researchers connected to the Clay Mathematics Institute and the Simons Foundation. The chapter explores base change, Beck–Chevalley conditions, and effective epimorphisms with connections to the descent perspectives in the work of Alexander Beilinson and Vladimir Drinfeld.

Examples and Constructions

The book supplies a range of examples: ∞-categories of spaces modeled on Kan complexes and Simplicial sets studied by Ken Brown and André Joyal, derived ∞-categories influenced by Alexei Bondal and Maxim Kontsevich, and ∞-topoi arising from sheaves on sites such as the étale site of schemes considered in contexts influenced by Alexander Grothendieck and Jean-Pierre Serre. Lurie gives constructions of Postnikov towers, localizations, and recollements echoing techniques used by Daniel Quillen and John Milnor, and connects to examples used in seminars at Massachusetts Institute of Technology and Stanford University.

Applications in Algebraic Geometry and Homotopy Theory

Higher Topos Theory underpins derived algebraic geometry and influences developments by Jacob Lurie himself and collaborators such as Dennis Gaitsgory, Vladimir Drinfeld, Bertrand Toën, Gabriele Vezzosi, Maxim Kontsevich, and Bertrand Toën’s school, interfacing with the moduli problems studied at Institute for Advanced Study and Clay Mathematics Institute programs. It provides foundations for spectral algebraic geometry related to concepts in Morava K-theory investigations by Jack Morava and chromatic homotopy theory from researchers at University of Chicago and Princeton University. The framework has been applied to the formulation of extended topological quantum field theories connected to work by Jacob Lurie at the Institute for Advanced Study and to interactions with categorical representation theory pursued at Harvard University and University of California, Berkeley.

Category:Mathematics textbooks