Higher category theory

Higher category theory

Higher category theory studies structures generalizing categorys by allowing morphisms between morphisms and further iterated cells; it grew from interactions among researchers associated with institutions like Institut des Hautes Études Scientifiques, École Normale Supérieure (Paris), and University of California, Berkeley. Influential figures such as Alexander Grothendieck, William Lawvere, Saunders Mac Lane, and Daniel Quillen provided antecedents that connect to modern developments by Jacob Lurie, André Joyal, John Baez, and Ross Street.

Introduction

Higher category theory generalizes the notion of category by introducing k-morphisms for k ≥ 0 and coherent compositions; early work in this direction arose alongside research at University of Cambridge (UK), Duke University, and Massachusetts Institute of Technology. The subject unites themes from Algebraic topology, Homotopy theory, and Algebraic geometry, with cross-fertilization from programs at Stony Brook University and collaborations involving authors affiliated with Institute for Advanced Study. Foundational motivations include formalizing coherence phenomena observed in the work of Jean Bénabou, Pierre Deligne, and Grothendieck, and tools developed by Daniel Quillen and Michael Boardman.

Foundations and Models

Foundational approaches include strict and weak models such as strict n-categories, weak n-categories, (∞,1)-categories, and (∞,n)-categories; primary models are due to researchers at Université de Paris, Rutgers University, and University of Cambridge (UK). Notable formalisms include Simplicial set–based models advanced by André Joyal and Jacob Lurie, Quasi-category theory associated with Simpson, Carlos and Boardman, Segal space approaches developed in seminars at Institut Henri Poincaré, and the opetopic and multitopic frameworks by Ross Street and collaborators. Other foundational systems include Model category structures introduced by Daniel Quillen, Complete Segal space axioms refined by Charles Rezk, and Theta category constructions connected to work at ETH Zurich and University of Cambridge (UK). Categorical coherence theorems trace back to results by Saunders Mac Lane and later generalizations by John Baez, Crans, Alissa, and Trimble, Todd.

Examples and Constructions

Standard examples arise from iterated morphism categories like 2-categories studied by Jean Bénabou and bicategories introduced by Max Kelly and G. M. Kelly, monoidal categories explored by Drinfeld, Vladimir and Igor Frenkel, and n-groupoids connected to the Grothendieck homotopy hypothesis proposed by Alexander Grothendieck. Constructions include nerve functors developed in workshops at University of Chicago and homotopy coherent nerves formalized by Voevodsky, Vladimir-adjacent programs and by Jacob Lurie. Higher categorical enrichments appear in examples from Topological quantum field theory programs influenced by investigators at Princeton University and Perimeter Institute for Theoretical Physics, and in categorical stacks and derived categories studied by groups at Harvard University and IHÉS. Algebraic examples include higher monads and operads from schools led by J. Peter May and Bertrand Toën.

Applications and Connections

Applications permeate diverse mathematical and physical domains: in Algebraic geometry via derived and spectral algebraic geometry advanced by Jacob Lurie and Bertrand Toën, in Homotopy theory through stabilization techniques championed at University of Chicago, and in mathematical physics through constructions at CERN-affiliated collaborations and institutes such as Perimeter Institute for Theoretical Physics. Higher categorical methods underpin modern formulations of Topological quantum field theory influenced by Michael Atiyah and Graeme Segal, and influence representation-theoretic programs linked to Pierre Deligne and David Ben-Zvi. Interdisciplinary connections include relationships with Logic and type-theoretic foundations developed by teams at Carnegie Mellon University and Microsoft Research (notably homotopy type theory projects), and with computational approaches advanced by researchers at University of Edinburgh and Oxford University.

Research Developments and Open Problems

Active research directions include comparisons between models (e.g., equivalences between quasi-categories, complete Segal spaces, and model categories) pursued by groups at University of California, Berkeley and Princeton University, categorification programs promoted by investigators at University of Oxford and Imperial College London, and the extension of the homotopy hypothesis to higher dimensions as envisioned by Alexander Grothendieck. Open problems concern precise coherence theorems in the weak n-category setting explored by Jean Penon and Carlos Simpson, existence and uniqueness questions for various (∞,n)-categorical enhancements pursued by laboratories at Mathematical Sciences Research Institute and Banff International Research Station, and explicit computational tools for higher-categorical structures of interest to communities at Simons Foundation-funded projects. Experimental frontiers involve interactions with Quantum field theory conjectures and categorical enumerative geometry initiatives associated with Clay Mathematics Institute and collaborative seminars at IHÉS.

Category:Category theory