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

- Timeline to 1945: before the definitions
- 1945–1970
- 1971–1980
- 1981–1990
- 1991–2000
- 2001–present
- See also
- Notes
- References

- 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.

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

Year | Contributors | Event |
---|---|---|

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 Zermelo–Abraham 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 Brauer–Cecil 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 Gödel–Paul 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 Eilenberg–Saunders 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 Gelfand–Mark 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 Birkhoff–Øystein 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. |

Year | Contributors | Event |
---|---|---|

1945 | Saunders Mac Lane–Samuel Eilenberg | Start of category theory: axioms for categories, functors and natural transformations. |

1945 | Norman Steenrod–Samuel 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. |

1948 | Cartan seminar | Writes 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 Cartan–Samuel 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 Ehresmann–Jean 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) |

1963 | Frank 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 |

1964 | Barry Mitchell–Peter Freyd | Mitchell–Freyd embedding theorem: Every small abelian category admits an exact and full embedding into the category of (left) modules Mod_{R} over some ring R |

1964 | Rudolf Haag–Daniel 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 Artin–Alexander 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. |

1965 | Edgar Brown | Abstract homotopy categories: A proper framework for the study of homotopy theory of CW complexes |

1965 | Max Kelly | dg-categories |

1965 | Max Kelly–Samuel Eilenberg | Enriched category theory: Categories C enriched over a category V are categories with Hom-sets Hom_{C} 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 |

1967 | Simon 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 |

1967 | Peter Gabriel–Michel Zisman | Axiomatizes 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 hΔ and hTop (Δ the category of simplicial sets) |

1967 | Jean Bénabou | V-actegories: A category C with an action ⊗ :V × C → C 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 |

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 (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 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 (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 |

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 (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 |

1972 | John 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 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. - Ross Street definition: A bicategory such that
- small bicoproducts exist;
- each monad admits a Kleisli construction (analogous to the quotient of an equivalence relation in a topos);
- it is locally small-cocomplete; and
- 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 |

1972 | William 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: - the types are the objects of
*E* - terms of type
*X*in the variables*x*_{i}of type*X*_{i}are polynomial expressions φ(*x*_{1},...,*x*_{m}): 1→*X*in the arrows*x*_{i}: 1→*X*_{i}in*E* - formulas are terms of type Ω (arrows from types to Ω)
- connectives are induced from the internal Heyting algebra structure of Ω
- quantifiers bounded by types and applied to formulas are also treated
- 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 |

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 M^{R} 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 M^{Δ} 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 (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 - φ
_{k}:*X*_{k}→*X*_{1}×_{X0}... ×_{X0}*X*_{1}(induced by*X*(α_{i}):*X*_{k}→*X*_{1}) assigned to*X*
is a weak equivalence of simplicial sets for Segal categories are a weak form of S-categories, in which composition is only defined up to a coherent system of equivalences. |

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 |

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*:Sub_{C}(B)→Sub_{C}(A) has a left adjoint and a right adjoint. Sub_{C}(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 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(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) |

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 (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 M^{D} 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 |

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 |

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 | 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 |

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 (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 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 (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 Freyd–David Yetter | Constructs the (compact braided) 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 (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 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 | 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: nCob_{C}→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 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 R ∩ S, 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 Reshetikhin–Vladimir Turaev–Edward Witten | Reshetikhin–Turaev–Witten invariants of knots from modular tensor categories of representations of quantum groups. |

Year | Contributors | Event |
---|---|---|

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. |

1992 | Yves 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. |

1992 | John 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 - for all
*X*,*Y*in Ob(C) the Hom-sets Hom_{C}(*X*,*Y*) are finite-dimensional chain complexes of**Z**-graded modules - for all objects
*X*_{1}, ...,*X*_{n}in Ob(C) there is a family of linear composition maps (the higher compositions)
*m*_{n}: Hom_{C}(*X*_{0},*X*_{1}) ⊗ Hom_{C}(*X*_{1},*X*_{2}) ⊗ ... ⊗ Hom_{C}(*X*_{n−1},*X*_{n}) → Hom_{C}(*X*_{0},*X*_{n})
- of degree
*n*− 2 (homological grading convention is used) for*n*≥ 1
- of degree
*m*_{1}is the differential on the chain complex Hom_{C}(*X*,*Y*)*m*_{n}satisfy the quadratic*A*_{∞}-associativity equation for all*n*≥ 0.
Examples are the Fukaya category Fuk( When there are no higher maps (trivial homotopies) The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of |

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. |

1993 | Daniel Freed | A new view on TQFT using modular tensor categories that unifies three approaches to TQFT (modular tensor categories from path integrals). |

1994 | Francis Borceux | Handbook 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 c_{1}(X) = 0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(Fuk_{X}) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of D^{b}(Coh_{Y}) (the bounded derived category of coherent sheaves on Y). |

1994 | Louis Crane-Igor Frenkel | Hopf categories and construction of 4D TQFTs by them. |

1994 | John Fischer | Defines the 2-category of 2-knots (knotted surfaces). |

1995 | Bob Gordon-John Power-Ross Street | Tricategories and a corresponding coherence theorem: Every weak 3-category is equivalent to a Gray 3-category. |

1995 | Ross Street–Dominic Verity | Surface diagrams for tricategories. |

1995 | Louis Crane | Coins categorification leading to the categorical ladder. |

1995 | Sjoerd Crans | A 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 Baez–James Dolan | Outlined a program in which n-dimensional TQFTs are described as n-category representations. |

1995 | John Baez–James Dolan | Proposed n-dimensional deformation quantization. |

1995 | John Baez–James 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: nCat_{k}→nCat_{k+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. |

1995 | Valentin 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. |

1998 | Claudio 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. |

1998 | André Hirschowitz-Carlos Simpson | Give 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 Baez–James 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),O_{A}) where A is an abelian category and to it is associated a topological space Spec(A) together with a sheaf of rings O_{A} on it. In the case when A = QCoh(X) for X a scheme the pair (Spec(A),O_{A}) is naturally isomorphic to the scheme (X^{Zar},O_{X}) using the equivalence of categories QCoh(Spec(R)) = Mod_{R}. 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 Calabi—Yau variety of dimension d then D^{b}(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 Bernstein–Igor Frenkel–Mikhail 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. |

1999 | Moira 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 Morel | Constructs the homotopy category of schemes. |

1999 | Ronald Brown–George Janelidze | 2-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 Eliashberg–Alexander Givental–Helmut 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. |

2000 | Paul 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. |

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 (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. |

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" (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). |

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 | Mario Caccamo | Formal category theoretical expanded λ-calculus for categories. |

2004 | Francis 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. |

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 (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 Mod_{W}. 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: R^{op} ⊗ 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(V^{op})^{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). |

2005 | Peter Selinger | Coined 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 |

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. 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 is a Grothendieck (nCatn+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. |

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 |

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. Jacob Lurie later publishes the complete proof of the cobordism hypothesis (2010). |

2019 | Brendan Fong–David 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 |

**Alexander Grothendieck** was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.

**Category theory** is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Homological algebra** is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory.

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of *objects* and *arrows*, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

**Lists of mathematics topics** cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.

The **Mathematics Subject Classification** (**MSC**) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.

In mathematics, an **operad** is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an *algebra over * to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.

In mathematics, **higher category theory** is the part of category theory at a *higher order*, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.

In higher category theory, the concept of higher categorical structures, such as (∞-categories), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group, but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space.

In the theory of operads in algebra and algebraic topology, an **A _{∞}-operad** is a parameter space for a multiplication map that is homotopy coherently associative.

In the theory of operads in algebra and algebraic topology, an **E _{∞}-operad** is a parameter space for a multiplication map that is associative and commutative "up to all higher homotopies".

In mathematics, especially (higher) category theory, **higher-dimensional algebra** is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.

**Steve Shnider** is a retired professor of mathematics at Bar Ilan University. He received a PhD in Mathematics from Harvard University in 1972, under Shlomo Sternberg. His main interests are in the differential geometry of fiber bundles; algebraic methods in the theory of deformation of geometric structures; symplectic geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014.

**Noncommutative algebraic geometry** is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them.

**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.

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.

**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.

**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.

- nLab, just as a higher-dimensional Wikipedia, started in late 2008; see nLab
- Zhaohua Luo; Categorical geometry homepage
- John Baez, Aaron Lauda; A prehistory of n-categorical physics
- Ross Street; An Australian conspectus of higher categories
- Elaine Landry, Jean-Pierre Marquis; Categories in context: historical, foundational, and philosophical
- Jim Stasheff; A survey of cohomological physics
- John Bell; The development of categorical logic
- Jean Dieudonné; The historical development of algebraic geometry
- Charles Weibel; History of homological algebra
- Peter Johnstone; The point of pointless topology
- Stasheff, Jim (January 21, 1996). "The Pre-History Of Operads". In Loday, Jean-Louis; Stasheff, James D.; Voronov, Alexander A. (eds.).
*Operads: Proceedings of Renaissance Conferences*. Contemporary Mathematics. Vol. 202. Providence, Rhode Island: American Mathematical Society. pp. 9–14. CiteSeerX 10.1.1.25.5089 . doi:10.1090/conm/202/02592. ISBN 0-8218-0513-4. ISSN 0271-4132. LCCN 96-37049. MR 1436913 . Retrieved 2021-12-08. - George Whitehead; Fifty years of homotopy theory
- Haynes Miller; The origin of sheaf theory

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