homotopy type theory

homotopy type theory
Presented at https://github.com/HoTT/book · CC BY-SA 3.0 · source
Name	Homotopy Type Theory
Field	Mathematical logic; Algebraic topology; Category theory
Introduced	2000s
Developers	Vladimir Voevodsky; Steve Awodey; Michael Warren; Peter Aczel; Thierry Coquand; Jeremy Avigad
Notable works	"Homotopy Type Theory: Univalent Foundations of Mathematics" (2013)

Contents

homotopy type theory

Homotopy type theory is an approach to the foundations of mathematics that synthesizes ideas from Algebraic topology, Category theory, Mathematical logic, Set theory, Type theory (computer science), and Constructive mathematics to reinterpret types as homotopical objects; it emerged through collaborations among researchers associated with institutions such as Institute for Advanced Study, Princeton University, Carnegie Mellon University, Université Paris-Saclay, and University of Gothenburg. The program attracted contributors linked to awards and projects including the Fields Medal recipient Vladimir Voevodsky and research groups at Microsoft Research, Institute for Advanced Study, MPI MiS, and initiatives related to the Univalent Foundations Project. Homotopy type theory has influenced work in formal proof systems at organizations like Cornell University, University of Pennsylvania, and INRIA.

Introduction

Homotopy type theory reinterprets constructive Type theory (computer science) in a homotopical and categorical framework influenced by results from Homotopy theory, Higher category theory, Model category ideas and geometric reasoning present in the work of mathematicians at University of Cambridge, Princeton University, Harvard University, Stanford University, and ETH Zurich. The subject synthesizes techniques from pioneers associated with Alexander Grothendieck-inspired programs, the Homotopy hypothesis discussions from seminars at IHÉS, and formalization efforts linked to Coq and Agda proof assistants developed by teams at INRIA and Chalmers University of Technology.

Foundational concepts draw on notions developed by researchers tied to Per Martin-Löf's constructive type theory and extensions influenced by William Lawvere's category-theoretic insights, Grothendieck’s higher stacks work, and categorical formulations from Saunders Mac Lane and Samuel Eilenberg. Central ideas include identity types interpreted via path spaces studied by authors connected to J. Peter May and Daniel Quillen, dependent types related to fibrations in the spirit of Quillen model categories, and equivalences treated through analogues to concepts from Eilenberg–Mac Lane spaces and André Joyal's quasi-category theory developed at institutions like Université de Montréal and Université Paris Diderot. The syntax-semantics interplay echoes frameworks used by researchers at Carnegie Mellon University, University of Edinburgh, and University of California, Berkeley.

The Univalence Axiom, proposed in the orbit of work by Vladimir Voevodsky and collaborators at Institute for Advanced Study and Princeton University, identifies equivalence with equality in a manner resonant with principles from Évariste Galois-style symmetry thinking and categorical equivalence concepts exploited in Category theory research at University of Chicago and Columbia University. Higher inductive types generalize constructors to include path constructors reminiscent of constructions in Algebraic topology and Homotopy theory studied by mathematicians at University of Oxford and École Normale Supérieure, enabling direct definitions of spheres, torii, and quotient spaces within type-theoretic syntax—methods influenced by seminars at IHÉS and affiliated with work from Max Planck Institute for Mathematics researchers.

Semantic models draw on classical model structures such as those of Daniel Quillen and advances in ∞-category models developed by researchers from Institut des Hautes Études Scientifiques, Kappa Research, Cornell University, and Perimeter Institute. Simplicial set models connect to research of André Joyal and Jacob Lurie at Institute for Advanced Study and Harvard University, while cubical models emerged from efforts at Microsoft Research and collaborators including groups at University of Illinois Urbana–Champaign and University of Birmingham. Constructive and sheaf-theoretic models relate to names associated with Grothendieck and schools at Université de Paris and University of Cambridge.

Applications and cross-connections appear in formalized mathematics projects at Princeton University, Carnegie Mellon University, University of Cambridge, and University of Oxford, influencing the design of proof assistants such as Coq and Agda and inspiring work in categorical logic at MIT and Caltech. Research ties link to computational complexity questions at University of Toronto and to constructive algebra programs linked to University of Illinois and Rutgers University. Interdisciplinary dialogues involve researchers from Microsoft Research, Google Research, and academic labs at ETH Zurich and Tsinghua University engaged in mechanized verification, homotopical algebra, and foundations reform discussions evoking institutions like Wissenschaftskolleg zu Berlin.

The development has been advanced using proof assistants associated with projects at INRIA and Microsoft Research; key tools include Coq (with contributors from École Polytechnique and INRIA), Agda (origins at Chalmers University of Technology), and cubical type theory implementations from teams at Microsoft Research and Carnegie Mellon University. Collaborative manuscripts such as the Univalent Foundations book were produced by groups linked to Institute for Advanced Study, University of Nottingham, and Harvard University, supported by workshops at Banff International Research Station and conferences at ICMS and CMU.

Critiques and open problems have been voiced by specialists at Princeton University, Cambridge University Press-affiliated scholars, and critics from research groups at University of Vienna and ETH Zurich concerning formal expressivity relative to classical Zermelo–Fraenkel approaches associated with Ernst Zermelo and Abraham Fraenkel, computational behavior in large-scale libraries maintained at University of Pennsylvania and Cornell University, and the generality of models compared to frameworks developed by Daniel Quillen and Jacob Lurie. Outstanding problems include foundations for synthetic constructions pursued by teams at IHÉS, coherence and computational interpretations advanced by researchers at Microsoft Research and Carnegie Mellon University, and the integration of homotopical foundations with mainstream curricula championed by educators at University of Oxford and Harvard University.