Timeline of category theory and related mathematics
This is a timeline of category theory and related mathematics. Its scope is taken as:
- Categories of abstract algebraic structures including representation theory and universal algebra;
- Homological algebra;
- Homotopical algebra;
- Topology using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology;
- Categorical logic and set theory in the categorical context such as algebraic set theory;
- Foundations of mathematics building on categories, for instance topos theory;
- Abstract geometry, including algebraic geometry, categorical noncommutative geometry, etc.
- Quantization related to category theory, in particular categorical quantization;
- Categorical physics relevant for mathematics.
Timeline to 1945: before the definitions
1945–1970
1971–1980
| Year | Contributors | Event |
| 1971 | Saunders Mac Lane | Influential book: Categories for the Working Mathematician, which became the standard reference in category theory |
| 1971 | Horst Herrlich–Oswald Wyler | Categorical topology: The study of topological categories of structured sets 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 Temperley–Elliott Lieb | Temperley–Lieb algebras: Algebras of tangles defined by generators of tangles and relations among them |
| 1971 | William Lawvere–Myles Tierney | Lawvere–Tierney topology on a topos |
| 1971 | William Lawvere–Myles Tierney | Topos theoretic forcing : 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 |
| 1971 | Bob 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 and coherence. A club is a special kind of 2-dimensional theory or a monoid in Cat/, each club giving a 2-monad on Cat |
| 1972 | John Isbell | Locales: A "generalized topological space" or "pointless spaces" defined by a 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 of a topos E is a topos and the functor f*:→ 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 Bénabou–Ross 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.
Jean Bénabou definition: A bicomplete symmetric monoidal closed category |
| 1972 | Peter May | 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. |
| 1972 | William Mitchell–Jean Bénabou | Mitchell–Bénabou internal language of a toposes: For a topos E with subobject classifier object Ω a language L the types are the objects of E 2) terms of type X in the variables xi of type Xi are polynomial expressions φ formulas are terms of type Ω 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 and ∈X. 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 |
| 1973 | Chris 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 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 Gersten | Shows 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 |
| 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 is a discrete simplicial set and the Segal map φk : Xk → X1 × X 0... × X 0X1 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 and an inflation x→I such that both P and I are in the category of pro/injective objects. A Frobenius category E is an example of a model category and the quotient E/P is its homotopy category hE |
| 1974 | Michael Artin | Generalizes Deligne–Mumford stacks to Artin stacks |
| 1974 | Robert 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 Chern–James Simons | Chern–Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D |
| 1975 | Saul Kripke–André Joyal | Kripke–Joyal semantics of the Mitchell–Bénabou internal language for toposes: The logic in categories of sheaves is first order intuitionistic predicate logic |
| 1975 | Radu 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 |
| 1975 | Manfred 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 |
| 1976 | Marcel 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:A→B in C the functor f*:SubC→SubC has a left adjoint and a right adjoint. SubC is the preorder of subobjects of A in C. Every topos is a logos. Heyting categories generalize Heyting algebras. |
| 1976 | Ross Street | Computads |
| 1977 | Michael Makkai–Gonzalo Reyes | Develops the Mitchell–Bénabou internal language of a topos thoroughly in a more general setting |
| 1977 | Andre Boileau–André Joyal–John Zangwill | LST 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 built out of a local theory S whose objects are the local sets and whose arrows are the local maps is a linguistic topos. Every topos E is equivalent to a linguistic topos C |
| 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 |
| 1978 | Francois 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 |
| 1978 | Don Anderson | Building on work of Kenneth Brown defines ABC 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 fibration category is canonically associated a 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 |
| 1979 | Don 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 Mebkhout | Proves the Riemann–Hilbert correspondence for complex manifolds |
| 1980 | Peter Freyd | Numerals in a topos |
1981–1990
| Year | Contributors | Event |
| 1981 | Shigeru Mukai | Mukai–Fourier transform |
| 1982 | Bob 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 | 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. Canonical modelizers are also used in pursuing stacks |
| 1983 | Alexander Grothendieck | Smooth functors and proper functors |
| 1984 | Vladimir 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 whose class form a topological category similar as universal algebra is for algebraic structures |
| 1984 | André Joyal | Simplicial sheaves. Simplicial sheaves on a topological space X is a model for the hypercomplete ∞-topos Sh^ |
| 1984 | André Joyal | Shows that the category of simplicial objects in a Grothendieck topos has a closed model structure |
| 1984 | André Joyal–Myles Tierney | Main Galois theorem for toposes: Every topos is equivalent to a category of étale presheaves on an open étale groupoid |
| 1985 | Michael Schlessinger–Jim Stasheff | L∞-algebras |
| 1985 | André Joyal–Ross Street | Braided monoidal categories |
| 1985 | André Joyal–Ross Street | Joyal–Street coherence theorem for braided monoidal categories |
| 1985 | Paul Ghez–Ricardo Lima–John Roberts | C*-categories |
| 1986 | Joachim Lambek–Phil Scott | Influential 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 which restricts to a dual equivalence of categories between corresponding full subcategories of sheaves and of étale bundles |
| 1986 | Peter Freyd–David Yetter | Constructs the monoidal category of tangles |
| 1986 | Vladimir Drinfeld–Michio 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 |
| 1987 | Jean-Yves Girard | Linear logic: The internal logic of a linear category |
| 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 Street–John Roberts | Formulates Street–Roberts conjecture: Strict ω-categories are equivalent to complicial sets |
| 1987 | André Joyal–Ross Street–Mei Chee Shum | Ribbon categories: A balanced rigid braided monoidal category |
| 1987 | Ross Street | n-computads |
| 1987 | Iain Aitchison | Bottom 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 | 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:nCobC→Hilb satisfying some axioms |
| 1988 | Edward Witten | Topological quantum field theory TQFT: A monoidal functor Z:nCob→Hilb satisfying some axioms |
| 1988 | Edward Witten | Topological string theory |
| 1989 | Hans Baues | Influential 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 and 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 R ∩ S, like in the category of sets with relations as morphisms for which a number of axioms are required. It generalizes the relation algebra to relations between different sorts. |
| 1990 | Nicolai Reshetikhin–Vladimir Turaev–Edward Witten | Reshetikhin–Turaev–Witten invariants of knots from modular tensor categories of representations of quantum groups. |
1991–2000
2001–present
| Year | Contributors | Event |
| 2001 | Charles Rezk | Constructs 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. |
| 2001 | Charles Rezk | Model toposes and their generalization homotopy toposes. |
| 2002 | Bertrand Toën-Gabriele Vezzosi | Segal toposes coming from Segal topologies, Segal sites and stacks over them. |
| 2002 | Bertrand 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". |
| 2002 | Peter Johnstone | Influential book: sketches of an elephant – a topos theory compendium. It serves as an encyclopedia of topos theory. |
| 2002 | Dennis Gaitsgory-Kari Vilonen-Edward Frenkel | Proves the geometric Langlands program for GL over finite fields. |
| 2003 | Denis-Charles Cisinski | Makes further work on ABC model categories and brings them back into light. From then they are called ABC model categories after their contributors. |
| 2004 | Dennis Gaitsgory | Extended the proof of the geometric Langlands program to include GL over C. This allows to consider curves over C instead of over finite fields in the geometric Langlands program. |
| 2004 | Mario Caccamo | Formal category theoretical expanded λ-calculus for categories. |
| 2004 | Francis Borceux-Dominique Bourn | Homological categories |
| 2004 | William Dwyer-Philips Hirschhorn-Daniel Kan-Jeffrey Smith | Introduces in the book: Homotopy limit functors on model categories and homotopical categories, a formalism of homotopical categories and homotopical functors that generalize the model category formalism of Daniel Quillen. A homotopical category has only a distinguished class of morphisms called weak equivalences and satisfy the two out of six axiom. This allow to define homotopical versions of initial and terminal objects, limit and colimit functors, 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 and all R, h and A lie in the autonomous monoidal bicategory Comodco of comonoids. Comod=Modcoop. 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 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 between classifying spaces of cohomology theories in the chromatic filtration and moduli spaces of supersymmetric QFTs parametrized by a manifold. |
| 2005 | Peter Selinger | 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 |
| 2006 | P. Carrasco-A.R. Garzon-E.M. Vitale | Categorical crossed modules |
| 2006 | Aslak Bakke Buan–Robert Marsh–Markus Reineke–Idun Reiten–Gordana 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. Luries 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 -category nCat is a Grothendieck -topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting. |
| 2006 | Marni Dee Sheppeard | Quantum toposes |
| 2007 | Bernhard 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 on the affine Grassmannian GrG = G)/G |
| 2008 | Ieke Moerdijk-Clemens Berger | Extends and improved the definition of Reedy category to become invariant under equivalence of categories. |
| 2008 | Michael J. Hopkins–Jacob Lurie | Sketch of proof of Baez-Dolan tangle hypothesis and Baez-Dolan cobordism hypothesis which classify extended TQFT in all dimensions. |