∞-category

∞-category
Name	∞-category
Field	Category theory, Homotopy theory, Higher algebra
Introduced	20th century
Notable	Jacob Lurie, André Joyal, Charles Rezk

∞-category

An ∞-category is a structure in higher category theory that generalizes the notion of a category by allowing morphisms, 2-morphisms, 3-morphisms, and so on, up to all higher dimensions, with coherent composition laws. It provides a unifying framework for homotopy-theoretic and higher-algebraic constructions used across modern mathematics, including algebraic topology, algebraic geometry, and mathematical physics. Researchers such as Jacob Lurie, André Joyal, Charles Rezk, Daniel Quillen, and William Thurston have contributed to its formalization and applications.

Introduction

The concept of an ∞-category arose from efforts to encode categories enriched over homotopical or higher-dimensional data, influenced by ideas from Alexander Grothendieck, Daniel Quillen, Henri Cartan, Jean-Pierre Serre, and Samuel Eilenberg. Early motivations linked to the study of derived functors in Derived Categories, model structures in Daniel Quillen's model categories, and coherence problems addressed by Saunders Mac Lane. Contemporary treatments often reference works by Jacob Lurie, André Joyal, and Charles Rezk and interact with frameworks developed by Michael Boardman, J. Peter May, John Milnor, and Dennis Sullivan.

Definitions and models

Multiple equivalent models capture the notion, each developed by prominent figures: Joyal's quasi-categories (simplicial sets) influenced by André Joyal; complete Segal spaces introduced by Charles Rezk; model category approaches stemming from Daniel Quillen; simplicial categories elaborated by J. P. May and Dwyer and Kan; and operadic and multicategorical formulations used by Jacob Lurie and Bertrand Toën. Other formalizations include stratified simplicial sets linked to John Baez's higher categories, complicial sets related to Ross Street, and Theta-spaces related to Carlos Simpson. Foundational technical tools draw on work by Alexander Grothendieck in homotopical algebra, Grothendieck's Pursuing Stacks program, Vladimir Voevodsky's homotopy type theory, and model-theoretic structures developed by Mark Hovey.

Basic properties and notions

Key notions include equivalences, limits, colimits, adjunctions, and monoidal structures adapted to higher morphisms, with classical inputs from Saunders Mac Lane and Max Kelly. Homotopy-coherent diagrams and mapping spaces relate to constructions by Daniel Quillen and J. P. May. Localizations and derived functors build on Grothendieck's techniques in homological algebra. Notions of presentability, compact generation, and accessibility connect to work by Peter Gabriel, Francois-Xavier Malgrange, and Henri Cartan. Higher categorical Yoneda lemmas and representability theorems echo results originating with Niels Henrik Abel-era influences and the modern categorical formulations of Samuel Eilenberg and Saunders Mac Lane.

Constructions and examples

Standard examples include the ∞-category of spaces (homotopy types) linked to Henri Poincaré's and Henri Cartan's topological studies; derived ∞-categories of rings and schemes related to Alexander Grothendieck and Jean-Pierre Serre; stable ∞-categories appearing in Verdier-style derived contexts and influenced by Jean-Louis Verdier; categories of spectra developed from work by J. H. C. Whitehead, G. W. Whitehead, and Frank Adams; and higher categories of operads used in May-style loop space recognition theorems. Constructions such as homotopy limits and colimits follow traditions from Daniel Quillen and J. P. May; model categorical presentations often cite Mark Hovey and Stefan Schwede. Examples from mathematical physics connect to approaches by Edward Witten, Cumrun Vafa, and Graeme Segal.

Applications and relations to other areas

∞-categories permeate modern algebraic geometry via derived and spectral algebraic geometry developed by Jacob Lurie and Bertrand Toën, influencing moduli problems studied by Pierre Deligne, Aise Johan de Jong, and Maxim Kontsevich. In topology, they underpin stable homotopy theory shaped by J. P. May, Frank Adams, and Douglas Ravenel. Homotopical aspects intersect with homological mirror symmetry frameworks proposed by Maxim Kontsevich and Paul Seidel. Interactions with mathematical logic and type theory involve Vladimir Voevodsky's univalent foundations and Per Martin-Löf's type theory. Connections to quantum field theory and string theory draw on work by Edward Witten, Nathan Seiberg, and Graeme Segal. Representation-theoretic and categorical geometric applications reach into areas shaped by George Lusztig, Peter Higgs, and David Kazhdan.

Historical development and key contributors

The roots lie in mid-20th-century algebraic topology and category theory with contributions by Alexander Grothendieck, Samuel Eilenberg, Saunders Mac Lane, and Daniel Quillen. Systematic higher-categorical formalisms advanced through the late 20th and early 21st centuries by André Joyal, Charles Rezk, Jacob Lurie, J. P. May, Mark Hovey, Vladimir Voevodsky, Ross Street, John Baez, Carlos Simpson, Bertrand Toën, and Jacob Lurie's students and collaborators. Key milestones include Quillen's model categories, Joyal's quasi-categories, Rezk's complete Segal spaces, and Lurie's consolidation in higher algebra and spectral algebraic geometry, influencing ongoing research at institutions like Institute for Advanced Study, Massachusetts Institute of Technology, Princeton University, and University of Cambridge.

Category:Category theory