1991–2000
| 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 m1 and m2 will be chain maps but the compositions mi of higher order are not chain maps; nevertheless they are Massey products. In particular it is a linear category. Examples are the Fukaya category Fuk(X) and loop space ΩX where X is a topological space and A∞-algebras as A∞-categories with one object. When there are no higher maps (trivial homotopies) C is a dg-category. Every A∞-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology. The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of A∞-categories and A∞-functors. Many features of A∞-categories and A∞-functors come from the fact that they form a symmetric closed multicategory, which is revealed in the language of comonads. From a higher dimensional perspective A∞-categories are weak ω-categories with all morphisms invertible. A∞-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects. |
| 1993 | John Barret-Bruce Westbury | Spherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane. |
| 1993 | Maxim Kontsevich | Kontsevich invariants for knots (are perturbation expansion Feynman integrals for the Witten functional integral) defined by the Kontsevich integral. They are the universal Vassiliev invariants for knots. |
| 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 c1(X)=0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(FukX) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of Db(CohY) (the bounded derived category of coherent sheaves on Y). |
| 1994 | Louis Crane-Igor Frenkel | Hopf categories and construction of 4D TQFTs by them. |
| 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 algebra) 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:nCatk→nCatk+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),OA) where A is an abelian category and to it is associated a topological space Spec(A) together with a sheaf of rings OA on it. In the case when A = QCoh(X) for X a scheme the pair (Spec(A),OA) is naturally isomorphic to the scheme (XZar,OX) using the equivalence of categories QCoh(Spec(R))=ModR. More generally abelian categories or triangulated categories or dg-categories or A∞-categories should be regarded as categories of quasicoherent sheaves (or complexes of sheaves) on noncommutative schemes. This is a starting point in noncommutative algebraic geometry. It means that one can think of the category A itself as a space. Since A is abelian it allows to naturally do homological algebra on noncommutative schemes and hence sheaf cohomology. |
| 1998 | Maxim Kontsevich | Calabi–Yau categories: A linear category with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map. If X is a smooth projective Calabi—Yau variety of dimension d then Db(Coh(X)) is a unital Calabi–Yau A∞-category of Calabi–Yau dimension d. A Calabi–Yau category with one object is a Frobenius algebra. |
| 1999 | Joseph 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 | 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. |
Read more about this topic: Timeline Of Category Theory And Related Mathematics