Timeline of category theory and related mathematics

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This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as:


In this article, and in category theory in general, ∞ = ω.

Timeline to 1945: before the definitions

1890 David Hilbert Resolution of modules and free resolution of modules.
1890 David Hilbert Hilbert's syzygy theorem is a prototype for a concept of dimension in homological algebra.
1893 David Hilbert A fundamental theorem in algebraic geometry, the Hilbert Nullstellensatz. It was later reformulated to: the category of affine varieties over a field k is equivalent to the dual of the category of reduced finitely generated (commutative) k-algebras.
1894 Henri Poincaré Fundamental group of a topological space.
1895 Henri Poincaré Simplicial homology.
1895 Henri Poincaré Fundamental work Analysis situs , the beginning of algebraic topology.
c.1910 L. E. J. Brouwer Brouwer develops intuitionism as a contribution to foundational debate in the period roughly 1910 to 1930 on mathematics, with intuitionistic logic a by-product of an increasingly sterile discussion on formalism.
1923 Hermann Künneth Künneth formula for homology of product of spaces.
1926 Heinrich Brandt defines the notion of groupoid.
1928 Arend Heyting Brouwer's intuitionistic logic made into formal mathematics, as logic in which the Heyting algebra replaces the Boolean algebra.
1929 Walther Mayer Chain complexes.
1930 Ernst ZermeloAbraham Fraenkel Statement of the definitive ZF-axioms of set theory, first stated in 1908 and improved upon since then.
c.1930 Emmy Noether Module theory is developed by Noether and her students, and algebraic topology starts to be properly founded in abstract algebra rather than by ad hoc arguments.
1932 Eduard Čech Čech cohomology, homotopy groups of a topological space.
1933 Solomon Lefschetz Singular homology of topological spaces.
1934 Reinhold Baer Ext groups, Ext functor (for abelian groups and with different notation).
1935 Witold Hurewicz Higher homotopy groups of a topological space.
1936 Marshall Stone Stone representation theorem for Boolean algebras initiates various Stone dualities.
1937 Richard BrauerCecil Nesbitt Frobenius algebras.
1938 Hassler Whitney "Modern" definition of cohomology, summarizing the work since James Alexander and Andrey Kolmogorov first defined cochains.
1940 Reinhold Baer Injective modules.
1940 Kurt GödelPaul Bernays Proper classes in set theory.
1940 Heinz Hopf Hopf algebras.
1941 Witold Hurewicz First fundamental theorem of homological algebra: Given a short exact sequence of spaces there exist a connecting homomorphism such that the long sequence of cohomology groups of the spaces is exact.
1942 Samuel EilenbergSaunders Mac Lane Universal coefficient theorem for Čech cohomology; later this became the general universal coefficient theorem. The notations Hom and Ext first appear in their paper.
1943 Norman Steenrod Homology with local coefficients.
1943 Israel GelfandMark Naimark Gelfand–Naimark theorem (sometimes called Gelfand isomorphism theorem): The category Haus of locally compact Hausdorff spaces with continuous proper maps as morphisms is equivalent to the category C*Alg of commutative C*-algebras with proper *-homomorphisms as morphisms.
1944 Garrett BirkhoffØystein Ore Galois connections generalizing the Galois correspondence: a pair of adjoint functors between two categories that arise from partially ordered sets (in modern formulation).
1944 Samuel Eilenberg "Modern" definition of singular homology and singular cohomology.
1945 Beno Eckmann Defines the cohomology ring building on Heinz Hopf's work.


1945 Saunders Mac LaneSamuel Eilenberg Start of category theory: axioms for categories, functors and natural transformations.
1945 Norman SteenrodSamuel Eilenberg Eilenberg–Steenrod axioms for homology and cohomology.
1945 Jean Leray Starts sheaf theory: At this time a sheaf was a map that assigned a module or a ring to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its p-th cohomology group.
1945 Jean Leray Defines Sheaf cohomology using his new concept of sheaf.
1946 Jean Leray Invents spectral sequences as a method for iteratively approximating cohomology groups by previous approximate cohomology groups. In the limiting case it gives the sought cohomology groups.
1948Cartan seminarWrites up sheaf theory for the first time.
1948 A. L. Blakers Crossed complexes (called group systems by Blakers), after a suggestion of Samuel Eilenberg: A nonabelian generalization of chain complexes of abelian groups which are equivalent to strict ω-groupoids. They form a category Crs that has many satisfactory properties such as a monoidal structure.
1949 John Henry Whitehead Crossed modules.
1949 André Weil Formulates the Weil conjectures on remarkable relations between the cohomological structure of algebraic varieties over C and the diophantine structure of algebraic varieties over finite fields.
1950 Henri Cartan In the book Sheaf theory from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology with support in an axiomatic form and more.
1950 John Henry Whitehead Outlines algebraic homotopy program for describing, understanding and calculating homotopy types of spaces and homotopy classes of mappings
1950 Samuel Eilenberg–Joe Zilber Simplicial sets as a purely algebraic model of well behaved topological spaces. A simplicial set can also be seen as a presheaf on the simplex category. A category is a simplicial set such that the Segal maps are isomorphisms.
1951 Henri Cartan Modern definition of sheaf theory in which a sheaf is defined using open subsets instead of closed subsets of a topological space and all the open subsets are treated at once. A sheaf on a topological space X becomes a functor resembling a function defined locally on X, and taking values in sets, abelian groups, commutative rings, modules or generally in any category C. In fact Alexander Grothendieck later made a dictionary between sheaves and functions. Another interpretation of sheaves is as continuously varying sets (a generalization of abstract sets). Its purpose is to provide a unified approach to connect local and global properties of topological spaces and to classify the obstructions for passing from local objects to global objects on a topological space by pasting together the local pieces. The C-valued sheaves on a topological space and their homomorphisms form a category.
1952 William Massey Invents exact couples for calculating spectral sequences.
1953 Jean-Pierre Serre Serre C-theory and Serre subcategories.
1952 Nobuo Yoneda Yoneda publishes his famous lemma. Yondeda's Lemma allows one to consider objects in a (small) category as a presheaves. Yoneda lemma plays a critical role in the study of representable functors in algebraic geometry. For example, even though it is never mentioned explicitly, it is central to the ideas of Grothendieck's "Fondements de la Géométrie Algébrique".
1955 Jean-Pierre Serre Shows there is a 1−1 correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring (Serre–Swan theorem).
1955 Jean-Pierre Serre Coherent sheaf cohomology in algebraic geometry.
1956 Jean-Pierre Serre GAGA correspondence.
1956 Henri CartanSamuel Eilenberg Influential book: Homological Algebra, summarizing the state of the art in its topic at that time. The notation Tor n and Ext n, as well as the concepts of projective module, projective and injective resolution of a module, derived functor and hyperhomology appear in this book for the first time.
1956 Daniel Kan Simplicial homotopy theory also called categorical homotopy theory: A homotopy theory completely internal to the category of simplicial sets.
1957 Charles EhresmannJean Bénabou Pointless topology building on Marshall Stone's work.
1957 Alexander Grothendieck Abelian categories in homological algebra that combine exactness and linearity.
1957 Alexander Grothendieck Influential Tohoku paper rewrites homological algebra; proving Grothendieck duality (Serre duality for possibly singular algebraic varieties). He also showed that the conceptual basis for homological algebra over a ring also holds for linear objects varying as sheaves over a space.
1957 Alexander Grothendieck Grothendieck's relative point of view, S-schemes.
1957 Alexander Grothendieck Grothendieck–Hirzebruch–Riemann–Roch theorem for smooth ; the proof introduces K-theory.
1957 Daniel Kan Kan complexes: Simplicial sets (in which every horn has a filler) that are geometric models of simplicial ∞-groupoids. Kan complexes are also the fibrant (and cofibrant) objects of model categories of simplicial sets for which the fibrations are Kan fibrations.
1958 Alexander Grothendieck Starts new foundation of algebraic geometry by generalizing varieties and other spaces in algebraic geometry to scheme which have the structure of a category with open subsets as objects and restrictions as morphisms. form a category that is a Grothendieck topos, and to a scheme and even a stack one may associate a Zariski topos, an étale topos, a fppf topos, a fpqc topos, a Nisnevich topos, a flat topos, ... depending on the topology imposed on the scheme. The whole of algebraic geometry was categorized with time.
1958 Roger Godement Monads in category theory (then called standard constructions and triples). Monads generalize classical notions from universal algebra and can in this sense be thought of as an algebraic theory over a category: the theory of the category of T-algebras. An algebra for a monad subsumes and generalizes the notion of a model for an algebraic theory.
1958 Daniel Kan Daniel Kan introduces Adjoint functors. They are critical, for example, in the theory of sheaves.
1958 Daniel Kan Limits in category theory.
1958 Alexander Grothendieck Fibred categories.
1959 Bernard Dwork Proves the rationality part of the Weil conjectures (the first conjecture).
1959 Jean-Pierre Serre Algebraic K-theory launched by explicit analogy of ring theory with geometric cases.
1960 Alexander Grothendieck Fiber functors
1960 Daniel Kan Kan extensions
1960 Alexander Grothendieck Formal algebraic geometry and formal schemes
1960 Alexander Grothendieck Representable functors
1960 Alexander Grothendieck Categorizes Galois theory (Grothendieck's Galois theory)
1960 Alexander Grothendieck Descent theory: An idea extending the notion of gluing in topology to scheme to get around the brute equivalence relations. It also generalizes localization in topology
1961 Alexander Grothendieck Local cohomology. Introduced at a seminar in 1961 but the notes are published in 1967
1961 Jim Stasheff Associahedra later used in the definition of weak n-categories
1961 Richard Swan Shows there is a 1−1 correspondence between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C(X) of continuous functions on X (Serre–Swan theorem)
1963Frank Adams–Saunders Mac Lane PROP categories and PACT categories for higher homotopies. PROPs are categories for describing families of operations with any number of inputs and outputs. Operads are special PROPs with operations with only one output
1963 Alexander Grothendieck Étale topology, a special Grothendieck topology on
1963 Alexander Grothendieck Étale cohomology
1963 Alexander Grothendieck Grothendieck toposes, which are categories which are like universes (generalized spaces) of sets in which one can do mathematics
1963 William Lawvere Algebraic theories and algebraic categories
1963 William Lawvere Founds categorical logic, discovers internal logics of categories and recognizes its importance and introduces Lawvere theories. Essentially categorical logic is a lift of different logics to being internal logics of categories. Each kind of category with extra structure corresponds to a system of logic with its own inference rules. A Lawvere theory is an algebraic theory as a category with finite products and possessing a "generic algebra" (a generic group). The structures described by a Lawvere theory are models of the Lawvere theory
1963 Jean-Louis Verdier Triangulated categories and triangulated functors. Derived categories and derived functors are special cases of these
1963 Jim Stasheff A-algebras: dg-algebra analogs of topological monoids associative up to homotopy appearing in topology (i.e. H-spaces)
1963 Jean Giraud Giraud characterization theorem characterizing Grothendieck toposes as categories of sheaves over a small site
1963 Charles Ehresmann Internal category theory: Internalization of categories in a category V with pullbacks is replacing the category Set (same for classes instead of sets) by V in the definition of a category. Internalization is a way to rise the categorical dimension
1963 Charles Ehresmann Multiple categories and multiple functors
1963 Saunders Mac Lane Monoidal categories, also called tensor categories: Strict 2-categories with one object made by a relabelling trick to categories with a tensor product of objects that is secretly the composition of morphisms in the 2-category. There are several object in a monoidal category since the relabelling trick makes 2-morphisms of the 2-category to morphisms, morphisms of the 2-category to objects and forgets about the single object. In general a higher relabelling trick works for n-categories with one object to make general monoidal categories. The most common examples include: ribbon categories, braided tensor categories, spherical categories, compact closed categories, symmetric tensor categories, modular categories, autonomous categories, categories with duality
1963 Saunders Mac Lane Mac Lane coherence theorem for determining commutativity of diagrams in monoidal categories
1964 William Lawvere ETCS Elementary Theory of the Category of Sets: An axiomatization of the category of sets which is also the constant case of an elementary topos
1964Barry Mitchell–Peter Freyd Mitchell–Freyd embedding theorem: Every small abelian category admits an exact and full embedding into the category of (left) modules ModR over some ring R
1964 Rudolf HaagDaniel Kastler Algebraic quantum field theory after ideas of Irving Segal
1964 Alexander Grothendieck Topologizes categories axiomatically by imposing a Grothendieck topology on categories which are then called sites. The purpose of sites is to define coverings on them so sheaves over sites can be defined. The other "spaces" one can define sheaves for except topological spaces are locales
1964 Michael ArtinAlexander Grothendieck ℓ-adic cohomology, technical development in SGA4 of the long-anticipated Weil cohomology.
1964 Alexander Grothendieck Proves the Weil conjectures except the analogue of the Riemann hypothesis
1964 Alexander Grothendieck Six operations formalism in homological algebra; Rf*, f−1, Rf!, f!, ⊗L, RHom, and proof of its closedness
1964 Alexander Grothendieck Introduced in a letter to Jean-Pierre Serre conjectural motives to express the idea that there is a single universal cohomology theory underlying the various cohomology theories for algebraic varieties. According to Grothendieck's philosophy there should be a universal cohomology functor attaching a pure motive h(X) to each smooth projective variety X. When X is not smooth or projective h(X) must be replaced by a more general mixed motive which has a weight filtration whose quotients are pure motives. The category of motives (the categorical framework for the universal cohomology theory) may be used as an abstract substitute for singular cohomology (and rational cohomology) to compare, relate and unite "motivated" properties and parallel phenomena of the various cohomology theories and to detect topological structure of algebraic varieties. The categories of pure motives and of mixed motives are abelian tensor categories and the category of pure motives is also a Tannakian category. Categories of motives are made by replacing the category of varieties by a category with the same objects but whose morphisms are correspondences, modulo a suitable equivalence relation; different equivalences give different theories. Rational equivalence gives the category of Chow motives with Chow groups as morphisms which are in some sense universal. Every geometric cohomology theory is a functor on the category of motives. Each induced functor ρ:motives modulo numerical equivalence→graded Q-vector spaces is called a realization of the category of motives, the inverse functors are called improvements. Mixed motives explain phenomena in as diverse areas as: Hodge theory, algebraic K-theory, polylogarithms, regulator maps, automorphic forms, L-functions, ℓ-adic representations, trigonometric sums, homotopy of algebraic varieties, algebraic cycles, moduli spaces and thus has the potential of enriching each area and of unifying them all.
1965Edgar BrownAbstract homotopy categories: A proper framework for the study of homotopy theory of CW complexes
1965 Max Kelly dg-categories
1965 Max KellySamuel Eilenberg Enriched category theory: Categories C enriched over a category V are categories with Hom-sets HomC not just a set or class but with the structure of objects in the category V. Enrichment over V is a way to rise the categorical dimension
1965 Charles Ehresmann Defines both strict 2-categories and strict n-categories
1966 Alexander Grothendieck Crystals (a kind of sheaf used in crystalline cohomology)
1966 William Lawvere ETAC Elementary theory of abstract categories, first proposed axioms for Cat or category theory using first-order logic
1967 Jean Bénabou Bicategories (weak 2-categories) and weak 2-functors
1967 William Lawvere Founds synthetic differential geometry
1967Simon Kochen–Ernst Specker Kochen–Specker theorem in quantum mechanics
1967 Jean-Louis Verdier Defines derived categories and redefines derived functors in terms of derived categories
1967Peter Gabriel–Michel ZismanAxiomatizes simplicial homotopy theory
1967 Daniel Quillen Quillen model categories and Quillen model functors: A framework for doing homotopy theory in an axiomatic way in categories and an abstraction of homotopy categories in such a way that hC = C[W−1] where W−1 are the inverted weak equivalences of the Quillen model category C. Quillen model categories are homotopically complete and cocomplete, and come with a built-in Eckmann–Hilton duality
1967 Daniel Quillen Homotopical algebra (published as a book and also sometimes called noncommutative homological algebra): The study of various model categories and the interplay between fibrations, cofibrations and weak equivalences in arbitrary closed model categories
1967 Daniel Quillen Quillen axioms for homotopy theory in model categories
1967 Daniel Quillen First fundamental theorem of simplicial homotopy theory: The category of simplicial sets is a (proper) closed (simplicial) model category
1967 Daniel Quillen Second fundamental theorem of simplicial homotopy theory: The realization functor and the singular functor is an equivalence of categories and hTop (Δ the category of simplicial sets)
1967 Jean Bénabou V-actegories: A category C with an action ⊗ :V × CC which is associative and unital up to coherent isomorphism, for V a symmetric monoidal category. V-actegories can be seen as the categorification of R-modules over a commutative ring R
1968 Chen-Ning Yang-Rodney Baxter Yang–Baxter equation, later used as a relation in braided monoidal categories for crossings of braids
1968 Alexander Grothendieck Crystalline cohomology: A p-adic cohomology theory in characteristic p invented to fill the gap left by étale cohomology which is deficient in using mod p coefficients for this case. It is sometimes referred to by Grothendieck as the yoga of de Rham coefficients and Hodge coefficients since crystalline cohomology of a variety X in characteristic p is like de Rham cohomology mod p of X and there is an isomorphism between de Rham cohomology groups and Hodge cohomology groups of harmonic forms
1968 Alexander Grothendieck Grothendieck connection
1968 Alexander Grothendieck Formulates the standard conjectures on algebraic cycles
1968 Michael Artin Algebraic spaces in algebraic geometry as a generalization of scheme
1968 Charles Ehresmann Sketches: An alternative way of presenting a theory (which is categorical in character as opposed to linguistic) whose models are to study in appropriate categories. A sketch is a small category with a set of distinguished cones and a set of distinguished cocones satisfying some axioms. A model of a sketch is a set-valued functor transforming the distinguished cones into limit cones and the distinguished cocones into colimit cones. The categories of models of sketches are exactly the accessible categories
1968 Joachim Lambek Multicategories
1968-1972 Michael Boardman and Rainer Vogt (1968), Peter May (1972) Operads: An abstraction of the family of composable functions of several variables together with an action of permutation of variables. Operads can be seen as algebraic theories and algebras over operads are then models of the theories. Each operad gives a monad on Top. Multicategories with one object are operads. PROPs generalize operads to admit operations with several inputs and several outputs. Operads are used in defining opetopes, higher category theory, homotopy theory, homological algebra, algebraic geometry, string theory and many other areas.
1969 Max Kelly-Nobuo Yoneda Ends and coends
1969 Pierre Deligne-David Mumford Deligne–Mumford stacks as a generalization of scheme
1969 William Lawvere Doctrines (category theory), a doctrine is a monad on a 2-category
1970 William Lawvere-Myles Tierney Elementary topoi: Categories modeled after the category of sets which are like universes (generalized spaces) of sets in which one can do mathematics. One of many ways to define a topos is: a properly cartesian closed category with a subobject classifier. Every Grothendieck topos is an elementary topos
1970 John Conway Skein theory of knots: The computation of knot invariants by skein modules. Skein modules can be based on quantum invariants


1971 Saunders Mac Lane Influential book: Categories for the Working Mathematician, which became the standard reference in category theory
1971 Horst HerrlichOswald Wyler Categorical topology: The study of topological categories of structured sets (generalizations of topological spaces, uniform spaces and the various other spaces in topology) and relations between them, culminating in universal topology. General categorical topology study and uses structured sets in a topological category as general topology study and uses topological spaces. Algebraic categorical topology tries to apply the machinery of algebraic topology for topological spaces to structured sets in a topological category.
1971 Harold TemperleyElliott Lieb Temperley–Lieb algebras: Algebras of tangles defined by generators of tangles and relations among them
1971 William LawvereMyles Tierney Lawvere–Tierney topology on a topos
1971 William LawvereMyles Tierney Topos theoretic forcing (forcing in toposes): Categorization of the set theoretic forcing method to toposes for attempts to prove or disprove the continuum hypothesis, independence of the axiom of choice, etc. in toposes
1971Bob Walters–Ross Street Yoneda structures on 2-categories
1971 Roger Penrose String diagrams to manipulate morphisms in a monoidal category
1971 Jean Giraud Gerbes: Categorified principal bundles that are also special cases of stacks
1971 Joachim Lambek Generalizes the Haskell–Curry–William–Howard correspondence to a three way isomorphism between types, propositions and objects of a cartesian closed category
1972 Max Kelly Clubs (category theory) and coherence (category theory). A club is a special kind of 2-dimensional theory or a monoid in Cat/(category of finite sets and permutations P), each club giving a 2-monad on Cat
1972John Isbell Locales: A "generalized topological space" or "pointless spaces" defined by a lattice (a complete Heyting algebra also called a Brouwer lattice) just as for a topological space the open subsets form a lattice. If the lattice possess enough points it is a topological space. Locales are the main objects of pointless topology, the dual objects being frames. Both locales and frames form categories that are each other's opposite. Sheaves can be defined over locales. The other "spaces" one can define sheaves over are sites. Although locales were known earlier John Isbell first named them
1972 Ross Street Formal theory of monads: The theory of monads in 2-categories
1972 Peter Freyd Fundamental theorem of topos theory: Every slice category (E,Y) of a topos E is a topos and the functor f*: (E,X) → (E,Y) preserves exponentials and the subobject classifier object Ω and has a right and left adjoint functor
1972 Alexander Grothendieck Grothendieck universes for sets as part of foundations for categories
1972 Jean BénabouRoss Street Cosmoses which categorize universes: A cosmos is a generalized universe of 1-categories in which you can do category theory. When set theory is generalized to the study of a Grothendieck topos, the analogous generalization of category theory is the study of a cosmos.
  1. Ross Street definition: A bicategory such that
  2. small bicoproducts exist;
  3. each monad admits a Kleisli construction (analogous to the quotient of an equivalence relation in a topos);
  4. it is locally small-cocomplete; and
  5. there exists a small Cauchy generator.

Cosmoses are closed under dualization, parametrization and localization. Ross Street also introduces elementary cosmoses.

Jean Bénabou definition: A bicomplete symmetric monoidal closed category

1972William Mitchell–Jean Bénabou Mitchell–Bénabou internal language of a toposes: For a topos E with subobject classifier object Ω a language (or type theory) L(E) where:
  1. the types are the objects of E
  2. terms of type X in the variables xi of type Xi are polynomial expressions φ(x1,...,xm): 1→X in the arrows xi: 1→Xi in E
  3. formulas are terms of type Ω (arrows from types to Ω)
  4. connectives are induced from the internal Heyting algebra structure of Ω
  5. quantifiers bounded by types and applied to formulas are also treated
  6. for each type X there are also two binary relations =X (defined applying the diagonal map to the product term of the arguments) and ∈X (defined applying the evaluation map to the product of the term and the power term of the arguments).

A formula is true if the arrow which interprets it factor through the arrow true:1→Ω. The Mitchell-Bénabou internal language is a powerful way to describe various objects in a topos as if they were sets and hence is a way of making the topos into a generalized set theory, to write and prove statements in a topos using first order intuitionistic predicate logic, to consider toposes as type theories and to express properties of a topos. Any language L also generates a linguistic topos E(L)

1973Chris Reedy Reedy categories: Categories of "shapes" that can be used to do homotopy theory. A Reedy category is a category R equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape R. The most important consequence of a Reedy structure on R is the existence of a model structure on the functor category MR whenever M is a model category. Another advantage of the Reedy structure is that its cofibrations, fibrations and factorizations are explicit. In a Reedy category there is a notion of an injective and a surjective morphism such that any morphism can be factored uniquely as a surjection followed by an injection. Examples are the ordinal α considered as a poset and hence a category. The opposite R° of a Reedy category R is also a Reedy category. The simplex category Δ and more generally for any simplicial set X its category of simplices Δ/X is a Reedy category. The model structure on MΔ for a model category M is described in an unpublished manuscript by Chris Reedy
1973 Kenneth Brown–Stephen GerstenShows the existence of a global closed model structure on the category of simplicial sheaves on a topological space, with weak assumptions on the topological space
1973 Kenneth Brown Generalized sheaf cohomology of a topological space X with coefficients a sheaf on X with values in Kans category of spectra with some finiteness conditions. It generalizes generalized cohomology theory and sheaf cohomology with coefficients in a complex of abelian sheaves
1973 William Lawvere Finds that Cauchy completeness can be expressed for general enriched categories with the category of generalized metric spaces as a special case. Cauchy sequences become left adjoint modules and convergence become representability
1973 Jean Bénabou Distributors (also called modules, profunctors, directed bridges)
1973 Pierre Deligne Proves the last of the Weil conjectures, the analogue of the Riemann hypothesis
1973 Michael Boardman–Rainer Vogt Segal categories: Simplicial analogues of A -categories. They naturally generalize simplicial categories, in that they can be regarded as simplicial categories with composition only given up to homotopy.

Def: A simplicial space X such that X0 (the set of points) is a discrete simplicial set and the Segal map

φk : XkX1 × X0 ... × X0X1 (induced by Xi): XkX1) assigned to X

is a weak equivalence of simplicial sets for k ≥ 2.

Segal categories are a weak form of S-categories, in which composition is only defined up to a coherent system of equivalences.
Segal categories were defined one year later implicitly by Graeme Segal. They were named Segal categories first by William Dwyer–Daniel Kan–Jeffrey Smith 1989. In their famous book Homotopy invariant algebraic structures on topological spaces, J. Michael Boardman and Rainer Vogt called them quasi-categories. A quasi-category is a simplicial set satisfying the weak Kan condition, so quasi-categories are also called weak Kan complexes

1973 Daniel Quillen Frobenius categories: An exact category in which the classes of injective and projective objects coincide and for all objects x in the category there is a deflation P(x)→x (the projective cover of x) and an inflation x→I(x) (the injective hull of x) such that both P(x) and I(x) are in the category of pro/injective objects. A Frobenius category E is an example of a model category and the quotient E/P (P is the class of projective/injective objects) is its homotopy category hE
1974 Michael Artin Generalizes Deligne–Mumford stacks to Artin stacks
1974Robert Paré Paré monadicity theorem: E is a topos → E° is monadic over E
1974 Andy Magid Generalizes Grothendieck's Galois theory from groups to the case of rings using Galois groupoids
1974 Jean Bénabou Logic of fibred categories
1974 John Gray Gray categories with Gray tensor product
1974 Kenneth Brown Writes a very influential paper that defines Browns categories of fibrant objects and dually Brown categories of cofibrant objects
1974 Shiing-Shen ChernJames Simons Chern–Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D
1975 Saul KripkeAndré Joyal Kripke–Joyal semantics of the Mitchell–Bénabou internal language for toposes: The logic in categories of sheaves is first-order intuitionistic predicate logic
1975Radu Diaconescu Diaconescu theorem: The internal axiom of choice holds in a topos → the topos is a boolean topos. So in IZF the axiom of choice implies the law of excluded middle
1975Manfred Szabo Polycategories
1975 William Lawvere Observes that Deligne's theorem about enough points in a coherent topos implies the Gödel completeness theorem for first-order logic in that topos
1976 Alexander Grothendieck Schematic homotopy types
1976Marcel Crabbe Heyting categories also called logoses: Regular categories in which the subobjects of an object form a lattice, and in which each inverse image map has a right adjoint. More precisely a coherent category C such that for all morphisms f:AB in C the functor f*:SubC(B)→SubC(A) has a left adjoint and a right adjoint. SubC(A) is the preorder of subobjects of A (the full subcategory of C/A whose objects are subobjects of A) in C. Every topos is a logos. Heyting categories generalize Heyting algebras.
1976 Ross Street Computads
1977 Michael Makkai–Gonzalo ReyesDevelops the Mitchell–Bénabou internal language of a topos thoroughly in a more general setting
1977Andre Boileau–André Joyal–John ZangwillLST, local set theory: Local set theory is a typed set theory whose underlying logic is higher-order intuitionistic logic. It is a generalization of classical set theory, in which sets are replaced by terms of certain types. The category C(S) built out of a local theory S whose objects are the local sets (or S-sets) and whose arrows are the local maps (or S-maps) is a linguistic topos. Every topos E is equivalent to a linguistic topos C(S(E))
1977 John Roberts Introduces most general nonabelian cohomology of ω-categories with ω-categories as coefficients when he realized that general cohomology is about coloring simplices in ω-categories. There are two methods of constructing general nonabelian cohomology, as nonabelian sheaf cohomology in terms of descent for ω-category valued sheaves, and in terms of homotopical cohomology theory which realizes the cocycles. The two approaches are related by codescent
1978 John Roberts Complicial sets (simplicial sets with structure or enchantment)
1978Francois Bayen–Moshe Flato–Chris Fronsdal–André Lichnerowicz–Daniel Sternheimer Deformation quantization, later to be a part of categorical quantization
1978 André Joyal Combinatorial species in enumerative combinatorics
1978Don AndersonBuilding on work of Kenneth Brown defines ABC (co)fibration categories for doing homotopy theory and more general ABC model categories, but the theory lies dormant until 2003. Every Quillen model category is an ABC model category. A difference to Quillen model categories is that in ABC model categories fibrations and cofibrations are independent and that for an ABC model category MD is an ABC model category. To an ABC (co)fibration category is canonically associated a (left) right Heller derivator. Topological spaces with homotopy equivalences as weak equivalences, Hurewicz cofibrations as cofibrations and Hurewicz fibrations as fibrations form an ABC model category, the Hurewicz model structure on Top. Complexes of objects in an abelian category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations form an ABC precofibration category
1979Don Anderson Anderson axioms for homotopy theory in categories with a fraction functor
1980 Alexander Zamolodchikov Zamolodchikov equation also called tetrahedron equation
1980 Ross Street Bicategorical Yoneda lemma
1980 Masaki Kashiwara–Zoghman MebkhoutProves the Riemann–Hilbert correspondence for complex manifolds
1980 Peter Freyd Numerals in a topos


1981 Shigeru Mukai Mukai–Fourier transform
1982Bob Walters Enriched categories with bicategories as a base
1983 Alexander Grothendieck Pursuing stacks: Manuscript circulated from Bangor, written in English in response to a correspondence in English with Ronald Brown and Tim Porter, starting with a letter addressed to Daniel Quillen, developing mathematical visions in a 629 pages manuscript, a kind of diary, and to be published by the Société Mathématique de France, edited by G. Maltsiniotis.
1983 Alexander Grothendieck First appearance of strict ∞-categories in pursuing stacks, following a 1981 published definition by Ronald Brown and Philip J. Higgins.
1983 Alexander Grothendieck Fundamental infinity groupoid: A complete homotopy invariant Π(X) for CW-complexes X. The inverse functor is the geometric realization functor |.| and together they form an "equivalence" between the category of CW-complexes and the category of ω-groupoids
1983 Alexander Grothendieck Homotopy hypothesis: The homotopy category of CW-complexes is Quillen equivalent to a homotopy category of reasonable weak ∞-groupoids
1983 Alexander Grothendieck Grothendieck derivators: A model for homotopy theory similar to Quilen model categories but more satisfactory. Grothendieck derivators are dual to Heller derivators
1983 Alexander Grothendieck Elementary modelizers: Categories of presheaves that modelize homotopy types (thus generalizing the theory of simplicial sets). Canonical modelizers are also used in pursuing stacks
1983 Alexander Grothendieck Smooth functors and proper functors
1984Vladimir Bazhanov–Razumov Stroganov Bazhanov–Stroganov d-simplex equation generalizing the Yang–Baxter equation and the Zamolodchikov equation
1984 Horst Herrlich Universal topology in categorical topology: A unifying categorical approach to the different structured sets (topological structures such as topological spaces and uniform spaces) whose class form a topological category similar as universal algebra is for algebraic structures
1984 André Joyal Simplicial sheaves (sheaves with values in simplicial sets). Simplicial sheaves on a topological space X is a model for the hypercomplete ∞-topos Sh(X)^
1984 André Joyal Shows that the category of simplicial objects in a Grothendieck topos has a closed model structure
1984 André JoyalMyles Tierney Main Galois theorem for toposes: Every topos is equivalent to a category of étale presheaves on an open étale groupoid
1985Michael Schlessinger–Jim Stasheff L-algebras
1985 André JoyalRoss Street Braided monoidal categories
1985 André JoyalRoss Street Joyal–Street coherence theorem for braided monoidal categories
1985Paul Ghez–Ricardo Lima–John Roberts C*-categories
1986 Joachim Lambek–Phil ScottInfluential book: Introduction to higher-order categorical logic
1986 Joachim Lambek–Phil Scott Fundamental theorem of topology: The section-functor Γ and the germ-functor Λ establish a dual adjunction between the category of presheaves and the category of bundles (over the same topological space) which restricts to a dual equivalence of categories (or duality) between corresponding full subcategories of sheaves and of étale bundles
1986 Peter FreydDavid Yetter Constructs the (compact braided) monoidal category of tangles
1986 Vladimir DrinfeldMichio Jimbo Quantum groups: In other words, quasitriangular Hopf algebras. The point is that the categories of representations of quantum groups are tensor categories with extra structure. They are used in construction of quantum invariants of knots and links and low-dimensional manifolds, representation theory, q-deformation theory, CFT, integrable systems. The invariants are constructed from braided monoidal categories that are categories of representations of quantum groups. The underlying structure of a TQFT is a modular category of representations of a quantum group
1986 Saunders Mac Lane Mathematics, form and function (a foundation of mathematics)
1987 Jean-Yves Girard Linear logic: The internal logic of a linear category (an enriched category with its Hom-sets being linear spaces)
1987 Peter Freyd Freyd representation theorem for Grothendieck toposes
1987 Ross Street Definition of the nerve of a weak n-category and thus obtaining the first definition of weak n-category using simplices
1987 Ross StreetJohn Roberts Formulates Street–Roberts conjecture: Strict ω-categories are equivalent to complicial sets
1987 André JoyalRoss Street–Mei Chee Shum Ribbon categories: A balanced rigid braided monoidal category
1987 Ross Street n-computads
1987Iain AitchisonBottom up Pascal triangle algorithm for computing nonabelian n-cocycle conditions for nonabelian cohomology
1987 Vladimir Drinfeld-Gérard Laumon Formulates geometric Langlands program
1987 Vladimir Turaev Starts quantum topology by using quantum groups and R-matrices to giving an algebraic unification of most of the known knot polynomials. Especially important was Vaughan Jones and Edward Wittens work on the Jones polynomial
1988 Alex Heller Heller axioms for homotopy theory as a special abstract hyperfunctor. A feature of this approach is a very general localization
1988 Alex Heller Heller derivators, the dual of Grothendieck derivators
1988 Alex Heller Gives a global closed model structure on the category of simplicial presheaves. John Jardine has also given a model structure in the category of simplicial presheaves
1988 Gregory Moore-Nathan Seiberg Rational Conformal Field Theories lead to modular tensor categories
1988 Graeme Segal Elliptic objects: A functor that is a categorified version of a vector bundle equipped with a connection, it is a 2D parallel transport for strings
1988 Graeme Segal Conformal field theory CFT: A symmetric monoidal functor Z: nCobCHilb satisfying some axioms
1988 Edward Witten Topological quantum field theory TQFT: A monoidal functor Z: nCobHilb satisfying some axioms
1988 Edward Witten Topological string theory
1989Hans BauesInfluential book: Algebraic homotopy
1989 Michael Makkai-Robert Paré Accessible categories: Categories with a "good" set of generators allowing to manipulate large categories as if they were small categories, without the fear of encountering any set-theoretic paradoxes. Locally presentable categories are complete accessible categories. Accessible categories are the categories of models of sketches. The name comes from that these categories are accessible as models of sketches.
1989 Edward Witten Witten functional integral formalism and Witten invariants for manifolds.
1990 Peter Freyd Allegories: An abstraction of the category of sets with relations as morphisms, it bears the same resemblance to binary relations as categories do to functions and sets. It is a category in which one has in addition to composition a unary operation reciprocation R° and a partial binary operation intersection RS, like in the category of sets with relations as morphisms (instead of functions) for which a number of axioms are required. It generalizes the relation algebra to relations between different sorts.
1990 Nicolai ReshetikhinVladimir TuraevEdward Witten Reshetikhin–Turaev–Witten invariants of knots from modular tensor categories of representations of quantum groups.


1991 Jean-Yves Girard Polarization of linear logic.
1991 Ross Street Parity complexes. A parity complex generates a free ω-category.
1991 André Joyal-Ross Street Formalization of Penrose string diagrams to calculate with abstract tensors in various monoidal categories with extra structure. The calculus now depends on the connection with low-dimensional topology.
1991 Ross Street Definition of the descent strict ω-category of a cosimplicial strict ω-category.
1991 Ross Street Top down excision of extremals algorithm for computing nonabelian n-cocycle conditions for nonabelian cohomology.
1992Yves Diers Axiomatic categorical geometry using algebraic-geometric categories and algebraic-geometric functors.
1992 Saunders Mac Lane-Ieke Moerdijk Influential book: Sheaves in geometry and logic.
1992John Greenlees-Peter May Greenlees-May duality
1992 Vladimir Turaev Modular tensor categories. Special tensor categories that arise in constructing knot invariants, in constructing TQFTs and CFTs, as truncation (semisimple quotient) of the category of representations of a quantum group (at roots of unity), as categories of representations of weak Hopf algebras, as category of representations of a RCFT.
1992 Vladimir Turaev-Oleg Viro Turaev-Viro state sum models based on spherical categories (the first state sum models) and Turaev-Viro state sum invariants for 3-manifolds.
1992 Vladimir Turaev Shadow world of links: Shadows of links give shadow invariants of links by shadow state sums.
1993 Ruth Lawrence Extended TQFTs
1993 David Yetter-Louis Crane Crane-Yetter state sum models based on ribbon categories and Crane-Yetter state sum invariants for 4-manifolds.
1993 Kenji Fukaya A -categories and A -functors: Most commonly in homological algebra, a category with several compositions such that the first composition is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. (associative up to higher homotopy). A stands for associative.

Def: A category C such that

  1. for all X,Y in Ob(C) the Hom-sets HomC(X,Y) are finite-dimensional chain complexes of Z-graded modules
  2. for all objects X1, ..., Xn in Ob(C) there is a family of linear composition maps (the higher compositions)
mn : HomC(X0,X1) ⊗ HomC(X1,X2) ⊗ ... ⊗ HomC(Xn−1,Xn) → HomC(X0,Xn)
of degree n  2 (homological grading convention is used) for n  1
  1. m1 is the differential on the chain complex HomC(X,Y)
  2. mn satisfy the quadratic A-associativity equation for all n  0.

m1 and m2 will be chain maps but the compositions mi of higher order are not chain maps; nevertheless they are Massey products. In particular it is a linear category.

Examples are the Fukaya category Fuk(X) and loop space ΩX where X is a topological space and A -algebras as A-categories with one object.

When there are no higher maps (trivial homotopies) C is a dg-category. Every A-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology.

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of A-categories and A-functors. Many features of A-categories and A-functors come from the fact that they form a symmetric closed multicategory, which is revealed in the language of comonads. From a higher-dimensional perspective A-categories are weak ω-categories with all morphisms invertible. A-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.

1993 John Barret-Bruce Westbury Spherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane.
1993 Maxim Kontsevich Kontsevich invariants for knots (are perturbation expansion Feynman integrals for the Witten functional integral) defined by the Kontsevich integral. They are the universal Vassiliev invariants for knots.
1993Daniel FreedA new view on TQFT using modular tensor categories that unifies three approaches to TQFT (modular tensor categories from path integrals).
1994Francis BorceuxHandbook of Categorical Algebra (3 volumes).
1994 Jean Bénabou–Bruno Loiseau Orbitals in a topos.
1994 Maxim Kontsevich Formulates the homological mirror symmetry conjecture: X a compact symplectic manifold with first Chern class c1(X) = 0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(FukX) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of Db(CohY) (the bounded derived category of coherent sheaves on Y).
1994 Louis Crane-Igor Frenkel Hopf categories and construction of 4D TQFTs by them.
1994John FischerDefines the 2-category of 2-knots (knotted surfaces).
1995Bob Gordon-John Power-Ross Street Tricategories and a corresponding coherence theorem: Every weak 3-category is equivalent to a Gray 3-category.
1995 Ross StreetDominic Verity Surface diagrams for tricategories.
1995 Louis Crane Coins categorification leading to the categorical ladder.
1995Sjoerd CransA general procedure of transferring closed model structures on a category along adjoint functor pairs to another category.
1995 André Joyal-Ieke Moerdijk AST, Algebraic set theory: Also sometimes called categorical set theory. It was developed from 1988 by André Joyal and Ieke Moerdijk, and was first presented in detail as a book in 1995 by them. AST is a framework based on category theory to study and organize set theories and to construct models of set theories. The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded, ...), the various constructions of the cumulative hierarchy of sets, forcing models, sheaf models and realisability models. Instead of focusing on categories of sets AST focuses on categories of classes. The basic tool of AST is the notion of a category with class structure (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a universal object (a universe)) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of ZF-algebras (Zermelo-Fraenkel algebras) and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first-order logic of elementary set theory on the other. The subcategory of sets in a class category is an elementary topos and every elementary topos occurs as sets in a class category. The class category itself always embeds into the ideal completion of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory (BIST) that is logically complete with respect to class category models. Therefore, class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The ZF-axioms are nothing but a description of the free ZF-algebra just as the Peano axioms are a description of the free monoid on one generator. In this perspective the models of set theory are algebras for a suitably presented algebraic theory and many familiar set theoretic conditions (such as well-foundedness) are related to familiar algebraic conditions (such as freeness). Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes.
1995 Michael Makkai SFAM, Structuralist foundation of abstract mathematics. In SFAM the universe consists of higher-dimensional categories, functors are replaced by saturated anafunctors, sets are abstract sets, the formal logic for entities is FOLDS (first-order logic with dependent sorts) in which the identity relation is not given a priori by first-order axioms but derived from within a context.
1995 John Baez-James Dolan Opetopic sets (opetopes) based on operads. Weak n-categories are n-opetopic sets.
1995 John Baez-James Dolan Introduced the periodic table of mathematics which identifies k-tuply monoidal n-categories. It mirrors the table of homotopy groups of the spheres.
1995 John BaezJames Dolan Outlined a program in which n-dimensional TQFTs are described as n-category representations.
1995 John BaezJames Dolan Proposed n-dimensional deformation quantization.
1995 John BaezJames Dolan Tangle hypothesis: The n-category of framed n-tangles in n + k dimensions is (n + k)-equivalent to the free weak k-tuply monoidal n-category with duals on one object.
1995 John Baez-James Dolan Cobordism hypothesis (Extended TQFT hypothesis I): The n-category of which n-dimensional extended TQFTs are representations, nCob, is the free stable weak n-category with duals on one object.
1995 John Baez-James Dolan Stabilization hypothesis: After suspending a weak n-category n + 2 times, further suspensions have no essential effect. The suspension functor S: nCatknCatk+1 is an equivalence of categories for k = n + 2.
1995 John Baez-James Dolan Extended TQFT hypothesis II: An n-dimensional unitary extended TQFT is a weak n-functor, preserving all levels of duality, from the free stable weak n-category with duals on one object to nHilb.
1995Valentin Lychagin Categorical quantization
1995 Pierre Deligne-Vladimir Drinfeld-Maxim Kontsevich Derived algebraic geometry with derived schemes and derived moduli stacks. A program of doing algebraic geometry and especially moduli problems in the derived category of schemes or algebraic varieties instead of in their normal categories.
1997 Maxim Kontsevich Formal deformation quantization theorem: Every Poisson manifold admits a differentiable star product and they are classified up to equivalence by formal deformations of the Poisson structure.
1998Claudio Hermida-Michael-Makkai-John Power Multitopes, Multitopic sets.
1998 Carlos Simpson Simpson conjecture: Every weak ∞-category is equivalent to a ∞-category in which composition and exchange laws are strict and only the unit laws are allowed to hold weakly. It is proven for 1,2,3-categories with a single object.
1998André Hirschowitz-Carlos SimpsonGive a model category structure on the category of Segal categories. Segal categories are the fibrant-cofibrant objects and Segal maps are the weak equivalences. In fact they generalize the definition to that of a Segal n-category and give a model structure for Segal n-categories for any n ≥ 1.
1998 Chris Isham–Jeremy Butterfield Kochen–Specker theorem in topos theory of presheaves: The spectral presheaf (the presheaf that assigns to each operator its spectrum) has no global elements (global sections) but may have partial elements or local elements. A global element is the analogue for presheaves of the ordinary idea of an element of a set. This is equivalent in quantum theory to the spectrum of the C*-algebra of observables in a topos having no points.
1998 Richard Thomas Richard Thomas, a student of Simon Donaldson, introduces Donaldson–Thomas invariants which are systems of numerical invariants of complex oriented 3-manifolds X, analogous to Donaldson invariants in the theory of 4-manifolds. They are certain weighted Euler characteristics of the moduli space of sheaves on X and "count" Gieseker semistable coherent sheaves with fixed Chern character on X. Ideally the moduli spaces should be a critical sets of holomorphic Chern–Simons functions and the Donaldson–Thomas invariants should be the number of critical points of this function, counted correctly. Currently such holomorphic Chern–Simons functions exist at best locally.
1998 John Baez Spin foam models: A 2-dimensional cell complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams are functors between spin network categories. Any slice of a spin foam gives a spin network.
1998 John BaezJames Dolan Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure.
1998 Alexander Rosenberg Noncommutative schemes: The pair (Spec(A),OA) where A is an abelian category and to it is associated a topological space Spec(A) together with a sheaf of rings OA on it. In the case when A = QCoh(X) for X a scheme the pair (Spec(A),OA) is naturally isomorphic to the scheme (XZar,OX) using the equivalence of categories QCoh(Spec(R)) = ModR. More generally abelian categories or triangulated categories or dg-categories or A-categories should be regarded as categories of quasicoherent sheaves (or complexes of sheaves) on noncommutative schemes. This is a starting point in noncommutative algebraic geometry. It means that one can think of the category A itself as a space. Since A is abelian it allows to naturally do homological algebra on noncommutative schemes and hence sheaf cohomology.
1998 Maxim Kontsevich Calabi–Yau categories: A linear category with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map. If X is a smooth projective CalabiYau variety of dimension d then Db(Coh(X)) is a unital Calabi–Yau A -category of Calabi–Yau dimension d. A Calabi–Yau category with one object is a Frobenius algebra.
1999 Joseph BernsteinIgor FrenkelMikhail Khovanov Temperley–Lieb categories: Objects are enumerated by nonnegative integers. The set of homomorphisms from object n to object m is a free R-module with a basis over a ring R. R is given by the isotopy classes of systems of (|n| + |m|)/2 simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs |n| points on the bottom and |m| points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley–Lieb categories are categorized Temperley–Lieb algebras.
1999Moira Chas–Dennis Sullivan Constructs string topology by cohomology. This is string theory on general topological manifolds.
1999 Mikhail Khovanov Khovanov homology: A homology theory for knots such that the dimensions of the homology groups are the coefficients of the Jones polynomial of the knot.
1999 Vladimir Turaev Homotopy quantum field theory HQFT
1999 Vladimir Voevodsky–Fabien MorelConstructs the homotopy category of schemes.
1999 Ronald Brown–George Janelidze2-dimensional Galois theory
2000 Vladimir Voevodsky Gives two constructions of motivic cohomology of varieties, by model categories in homotopy theory and by a triangulated category of DM-motives.
2000 Yasha EliashbergAlexander GiventalHelmut Hofer Symplectic field theory SFT: A functor Z from a geometric category of framed Hamiltonian structures and framed cobordisms between them to an algebraic category of certain differential D-modules and Fourier integral operators between them and satisfying some axioms.
2000Paul Taylor [1] ASD (Abstract Stone duality): A reaxiomatisation of the space and maps in general topology in terms of λ-calculus of computable continuous functions and predicates that is both constructive and computable. The topology on a space is treated not as a lattice, but as an exponential object of the same category as the original space, with an associated λ-calculus. Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role (an overt object is the dual to a compact object), forming an arithmetic universe (pretopos with lists) with general recursion.


2001Charles RezkConstructs a model category with certain generalized Segal categories as the fibrant objects, thus obtaining a model for a homotopy theory of homotopy theories. Complete Segal spaces are introduced at the same time.
2001Charles Rezk Model toposes and their generalization homotopy toposes (a model topos without the t-completeness assumption).
2002 Bertrand Toën-Gcabriele Vezzosi Segal toposes coming from Segal topologies, Segal sites and stacks over them.
2002Bertrand Toën-Gabriele Vezzosi Homotopical algebraic geometry: The main idea is to extend schemes by formally replacing the rings with any kind of "homotopy-ring-like object". More precisely this object is a commutative monoid in a symmetric monoidal category endowed with a notion of equivalences which are understood as "up-to-homotopy monoid" (e.g. E-rings).
2002 Peter Johnstone Influential book: sketches of an elephant – a topos theory compendium. It serves as an encyclopedia of topos theory (two out of three volumes published as of 2008).
2003Denis-Charles CisinskiMakes further work on ABC model categories and brings them back into light. From then they are called ABC model categories after their contributors.
2004Mario CaccamoFormal category theoretical expanded λ-calculus for categories.
2004Francis Borceux-Dominique Bourn Homological categories
2004 Samson Abramsky and Bob Coecke Paper A categorical semantics of quantum protocols published that starts the Oxford school of Categorical Quantum Mechanics, based on the theory of compact closed categories.
2004William Dwyer-Philips Hirschhorn-Daniel Kan-Jeffrey SmithIntroduces in the book Homotopy limit functors on model categories and homotopical categories a formalism of homotopical categories and homotopical functors (weak equivalence preserving functors) that generalize the model category formalism of Daniel Quillen. A homotopical category has only a distinguished class of morphisms (containing all isomorphisms) called weak equivalences and satisfy the two out of six axiom. This allows to define homotopical versions of initial and terminal objects, limit and colimit functors (that are computed by local constructions in the book), completeness and cocompleteness, adjunctions, Kan extensions and universal properties.
2004 Dominic Verity Proves the Street-Roberts conjecture.
2004 Ross Street Definition of the descent weak ω-category of a cosimplicial weak ω-category.
2004 Ross Street Characterization theorem for cosmoses: A bicategory M is a cosmos iff there exists a base bicategory W such that M is biequivalent to ModW. W can be taken to be any full subbicategory of M whose objects form a small Cauchy generator.
2004 Ross Street-Brian Day Quantum categories and quantum groupoids: A quantum category over a braided monoidal category V is an object R with an opmorphism h: Rop ⊗ R → A into a pseudomonoid A such that h* is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, h and A lie in the autonomous monoidal bicategory Comod(V)co of comonoids. Comod(V) = Mod(Vop)coop. Quantum categories were introduced to generalize Hopf algebroids and groupoids. A quantum groupoid is a Hopf algebra with several objects.
2004 Stephan Stolz-Peter Teichner Definition of nD QFT of degree p parametrized by a manifold.
2004 Stephan Stolz-Peter Teichner Graeme Segal proposed in the 1980s to provide a geometric construction of elliptic cohomology (the precursor to tmf) as some kind of moduli space of CFTs. Stephan Stolz and Peter Teichner continued and expanded these ideas in a program to construct TMF as a moduli space of supersymmetric Euclidean field theories. They conjectured a Stolz-Teichner picture (analogy) between classifying spaces of cohomology theories in the chromatic filtration (de Rham cohomology, K-theory, Morava K-theories) and moduli spaces of supersymmetric QFTs parametrized by a manifold (proved in 0D and 1D).
2005Peter SelingerCoined the term Dagger categories and dagger functors. Dagger categories seem to be part of a larger framework involving n-categories with duals.
2005 Peter Ozsváth-Zoltán Szabó Knot Floer homology
2006P. Carrasco-A.R. Garzon-E.M. Vitale Categorical crossed modules
2006Aslak Bakke Buan–Robert Marsh–Markus Reineke–Idun ReitenGordana Todorov Cluster categories: Cluster categories are a special case of triangulated Calabi–Yau categories of Calabi–Yau dimension 2 and a generalization of cluster algebras.
2006 Jacob Lurie Monumental book: Higher topos theory: In its 940 pages Jacob Lurie generalizes the common concepts of category theory to higher categories and defines n-toposes, ∞-toposes, sheaves of n-types, ∞-sites, ∞-Yoneda lemma and proves Lurie characterization theorem for higher-dimensional toposes. Lurie's theory of higher toposes can be interpreted as giving a good theory of sheaves taking values in ∞-categories. Roughly an ∞-topos is an ∞-category which looks like the ∞-category of all homotopy types. In a topos mathematics can be done. In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the (n+1)-category nCat is a Grothendieck (n+1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting. An introduction into this circle of ideas can be found in the Kerodon project.
2007Bernhard Keller-Thomas Hugh d-cluster categories
2007 Dennis Gaitsgory-Jacob Lurie Presents a derived version of the geometric Satake equivalence and formulates a geometric Langlands duality for quantum groups.

The geometric Satake equivalence realized the category of representations of the Langlands dual group LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian GrG = G((t))/G[[t]] of the original group G.

2008 Ieke Moerdijk-Clemens BergerExtends and improved the definition of Reedy category to become invariant under equivalence of categories.
2008 Michael J. HopkinsJacob Lurie Sketch of proof of Baez-Dolan tangle hypothesis and Baez-Dolan cobordism hypothesis which classify extended TQFT in all dimensions. Jacob Lurie later publishes the complete proof of the cobordism hypothesis (2010).
2019 Brendan FongDavid Spivak First textbook for the emerging field identifying itself as applied category theory, in which category theory is applied outside pure mathematics: An Invitation to Applied Category Theory: Seven Sketches in Compositionality

See also


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Ronald Brown FLSW is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.

<span class="mw-page-title-main">Cyclohedron</span>

In geometry, the cyclohedron is a -dimensional polytope where can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl and by Rodica Simion. Rodica Simion describes this polytope as an associahedron of type B.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

<span class="mw-page-title-main">Jim Stasheff</span> American mathematician

James Dillon Stasheff is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics.

<span class="mw-page-title-main">Alexander A. Voronov</span> Russian-American mathematician

Alexander A. Voronov is a Russian-American mathematician specializing in mathematical physics, algebraic topology, and algebraic geometry. He is currently a Professor of Mathematics at the University of Minnesota and a Visiting Senior Scientist at the Kavli Institute for the Physics and Mathematics of the Universe.