nLab Timeline of category theory and related mathematics

This is a timeline of category theory and related mathematics.

Timeline

Year		Contributors		Event
1848		Arthur Cayley[1]		Computations with complexes, Koszul resolutions, notion of exactness; main result: the resultant is a determinant of a Koszul complex.
1890		David Hilbert		resolution? and free resolution of modules
1890		David Hilbert		Hilbert syzygy theorem?
1893		David Hilbert		fundamental theorem of algebraic geometry also called 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
1923		Otto Künneth?		Künneth formula for (co)homology of product of spaces
1926		Otto Schreier		Classifies nonabelian extensions of groups having implicitly notions of a pseudofunctor and nonabelian cohomology in dimensions up to 3
1929		Walther Mayer?		chain complexes
1930		Ernst Zermelo–Abraham Fraenkel		Statement of the final ZF-axioms of set theory after being first stated in 1908 and improved upon since then
1932		[Georges de Rham]		de Rham theorem: For a smooth manifold the de Rham cohomology is isomorphic to the singular cohomology with coefficients in R.
1932		Eduard Čech		Čech cohomology, higher homotopy groups of a topological space though nobody paid attention because they were all abelian.
1933		Solomon Lefschetz		singular homology of a topological space
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
1940		Heinz Hopf		Hopf algebras
1941		Witold Hurewicz		first fundamental theorem of homological algebra?: Given a short exact sequence of spaces there exists 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
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: A sheaf on a topological space $X$ is a functor reminding one of 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 variable set?s (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
1945		Jean Leray		sheaf cohomology
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. The category of spectral sequences? is an abelian category
1948		Cartan seminar (query 4 down)		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 generalizations of chain complexes of abelian groups which are equivalent to strict ∞-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 $\mathbf{C}$ 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 (étalé 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
1951		M M Postnikov		publishes the results of his thesis: Postnikov system
1952		William Massey		invents exact couples for calculating spectral sequences
1953		Jean-Pierre Serre		Serre C-theory? and Serre subcategories
1955		Jean-Pierre Serre		shows there is a one-to-one correspondence between algebraic vector bundles over a noetherian affine variety and finitely generated projective modules over its coordinate ring (Serre-Swan theorem)
1955		Jean-Pierre Serre		coherent sheaf cohomology? in algebraic geometry
1955		Michel Lazard		Introduces “analyseurs”, a version of the future operads of Peter May
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		the Grothendieck relative point of view?, S-schemes
1957		Alexander Grothendieck		Grothendieck-Hirzebruch-Riemann-Roch theorem? for smooth schemes
1957		Daniel Kan		Kan complexes: simplicial sets (in which every horn has a filler) that are geometric models of ∞-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 schemes which have the structure of a category with open subsets as objects and restrictions as morphisms. Schemes 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 (which he called standard constructions). 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		adjoint functors
1958		Daniel Kan		limits in category theory
1958		Alexander Grothendieck		Introduces pseudofunctors and descent theory in FGA but publish them later with Pierre Gabriel in SGA1 1961 modernized into fibred categories.
1959		Alexander Grothendieck		Introduces formal algebraic geometry and formal schemes (partly with Pierre Cartier) in a seminar Bourbaki and publish it in FGA.
1959		Bernard Dwork		proves the rationality part of the Weil conjectures (the first conjecture)
1960		Alexander Grothendieck		fiber functors
1960		Daniel Kan		Kan extensions
1960		Alexander Grothendieck		representable functors
1960		Alexander Grothendieck		categorizes Galois theory (Grothendieck Galois theory)
1960		Alexander Grothendieck		descent theory: an idea extending the notion of gluing in topology to schemes to get around the brute equivalence relations. It also generalizes localization in topology
1960		Pierre Gabriel[2]		Reconstruction of a scheme from the category of quasicoherent sheaves over it (Gabriel–Rosenberg theorem in the separated quasicompact case and a precursor of noncommutative algebraic geometry) and abelian localization.
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 one-to-one 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		etale topology, a special Grothendieck topology on schemes
1963		Alexander Grothendieck		etale 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 their 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 (doctrine) 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” (such as a generic group). The structures described by a Lawvere theory are models of the Lawvere theory
1963		Jean-Louis Verdier		after the advice of Grothendieck, defined triangulated categories and triangulated functors including the main examples: derived categories. Studied derived functors in the triangualted setup
1963		Jim Stasheff		$A_\infty$-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 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: $2$-categories with one object made by a relabelling trick into categories with a tensor product of objects that is secretly the composition of morphisms in the $2$-category. There are several objects in a monoidal category since the relabelling trick makes $2$-morphisms of the $2$-category into morphisms, morphisms of the $2$-category into 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 Graeme 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		Alexander Grothendieck		$l$-adic cohomology?
1964		Alexander Grothendieck		proves the Weil conjectures except the analogue of the Riemann hypothesis
1964		Alexander Grothendieck		six operations formalism in homological algebra; $R f_*$, $f^{-1}$, $R f_!$, $f_!$, $\otimes^L$, $R Hom$, 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 memoirs this idea was born in 1958. 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 $\rho$ from motives modulo numerical equivalence to graded $\mathbf{Q}$-vector spaces is called a realization of the category of motives, the inverse functors are called improvement?s. 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, and 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 the 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 raise 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		follwing Grothendieck’s advice, defined triangulated categories and constructed derived categories; redefinition of derived functors in terms of triangulated categories
1967		Peter Gabriel–Michel Zisman		Famous book “Calculus of fractions and homotopy theory” sets a standard on the categorical approach to localization and axiomatizes simplicial homotopy theory.
1967		Daniel Quillen		Quillen model categories and Quillen model functor?s: A framework for doing homotopy theory in an axiomatic way in categories and an abstraction of homotopy categories in such a way that $h C = C[W^{-1}]$ where $W^{-1}$ consists of 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): introduces model categories in terms of fibrations, cofibrations and weak equivalences and studies main examples, 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? constitute an equivalence of categories $h \Delta$ and $h Top$ ($\Delta$ the category of simplicial sets)
1967		Jean Bénabou		$V$-actegories: a category $C$ with an action $\otimes: V \times C \to C$ 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 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 etale 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 schemes
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
1969		Max Kelly–Nobuo Yoneda		ends and coends
1969		Pierre Deligne–David Mumford		Deligne?Mumford stacks as a generalization of schemes
1969		William Lawvere		doctrines, a doctrine is a monad on a $2$-category
1970		William Lawvere-Myles Tierney		Elementary toposes: Categories modeled after the category of sets which are like universe?s (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 module?s. Skein modules can be based on quantum invariant?s
1970		Jean Bénabou-Jacques Roubaud		connect the descent in fibered categories with monadic descent: Benabou-Roubaud theorem
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 algebra?s: Algebras of tangle?s 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 diagram?s to manipulate morphisms in a monoidal category
1971		Jean Giraud?		Gerbe?s: 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 others 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		Universes for sets
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.
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. PROP?s 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 language of a 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 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 categegory of simplicial sheaves on a topological space, with weak asumptions 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		John 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.
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 Grothendiecks 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-Andre 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 Delignes 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 type?s
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(B)→SubC(A) has a left adjoint and a right adjoint. SubC(A) is the preorder of subobjects of A (the full subcategory of C/A whose objects are subobjects of A) in C. Every topos is a logos. Heyting categories generalize Heyting algebras.
1976		Ross Street		Computads?
1977		Peter Johnstone		Very influential book “Topos theory” (circulated as a preprint a year earlier).
1977		Michael Makkai-Gonzalo Reyes		Develops the Mitchell-Bénabou internal language? of a topos thoroughly in a more general setting
1977		Andre Boileau-Andre Joyal-Jon 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 ∞-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 set?s (simplicial sets with structure or enchantment)
1978		Francois Bayen-Moshe Flato-Chris Fronsdal-Andre Lichnerowicz-Daniel Sternheimer		Deformation quantization, later to be a part of categorical quantization
1978		Andre 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 MD is an ABC model category. To a 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
1978-1979		Alexander Beilinson[3]		Two articles on the structures of derived categories of coherent sheaves on projective spaces, which started a rich theory of relations between linear algebra of quivers and triangulated categories coming from algebraic geometry. This is continued in 1979 a famous related article of Bernstein-Gel’fand-Gel’fand with importance to physics and representation theory.
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		Shigeru Mukai		Mukai-Fourier transform?
1982		Bob Walters		Enriched categories? with bicategories as a base
1982		Martin Hyland		Devises the effective topos, an environment for recursive mathematics
1983		Alexander Grothendieck		Pursuing stacks: Correspondence by mail with Daniel Quillen about Alexander Grothendieck’s mathematical visions written down in a 629 pages manuscript
1983		Alexander Grothendieck		First appearance of strict ∞-categories in pursuing stacks
1983		Alexander Grothendieck		Fundamental infinity groupoid?: A complete homotopy invariant Π?(X) for CW-complexes X. The inverse functor is the geometric realization functor $\vert{.}\vert$ 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 modelizer?s: Categories of presheaves that modelize homotopy types (thus generalizing the theory of simplicial sets). Canonical modelizer?s are also used in pursuing stacks
1983		Alexander Grothendieck		Smooth functor?s 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		Andre 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		Andre Joyal		Shows that the category of simplicial objects in a Grothendieck topos has a closed model structure
1984		Andre 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		Andre Joyal-Ross Street		Braided monoidal categories?
1985		Andre 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 Drinfel'd-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 invariant?s 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-set?s 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 ∞-categories are equivalent to complicial sets
1987		Andre 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 Drinfel'd-Gerard 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 polynomial?s. 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 derivator?s, the dual of Grothendieck derivator?s
1988		Alex Heller		Gives a global closed model structure on the category of simplicial presheaves. John Jardine has also given a model structure for the category of simplicial presheaves
1988		Graeme Segal		Elliptic object?s: 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 invariant?s for manifolds
1990		Alexei Bondal-Mikhail Kapranov		Enhanced triangulated categories
1990		Peter Freyd		Allegories (category theory)?: 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 (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 invariant?s of knots from modular tensor categories of representations of quantum groups
1990		Cartier et al.		Grothendieck Festschrift in 3 volumes with historical contributions including Thomason-Troubaugh article on algebraic K-theory; Deligne: Categories Tannakiennes and Breen’s “Bitorseurs et cohomologie nonabeliennes”
1991		Jean-Yves Girard		Polarization? of linear logic
1991		Ross Street		Parity complex?es. A parity complex generates a free ∞-category
1991		Andre Joyal-Ross Street		Formalization of Penrose string diagrams to calculate with abstract tensor?s in various monoidal categories with extra structure. The calculus now depend 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 constructiong knot invariants, in constructing TQFT?s and CFT?s, 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 model?s 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 sum?s
1993		Paul Taylor		ASD (Abstract Stone Duality): A reaxiomatisation of the notions of space and map in general topology in terms of $\lambda$-calculus of computable continuous functions and predicates that is both constructive and computable
1993		Ruth Lawrence		Extended TQFTs
1993		David Yetter-Louis Crane		Crane-Yetter state sum model?s based on ribbon categories and Crane-Yetter state sum invariant?s 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.
1993		John Barrett?-Bruce Westbury		Spherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane
1993		Maxim Kontsevich		Kontsevich invariant?s 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 3 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 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 <=> 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-knot?s (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 which is a much simpler
1995		Ross Street-Dominic Verity		Surface diagram?s 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 started to be developed in 1988 by André Joyal and Ieke Moerdijk and was first presented in detail as a book in 1995 by them. AST is a robust framework based on category theory to study and organize set theories and to construct models of set theories?
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 set?s (opetopes) based on operads. Weak n-categories are n-opetopic sets
1995		John Baez-James Dolan		Introduces the periodic table of k-tuply monoidal n-categories.
1995		John Baez-James Dolan		Outlines a program in which n-dimensional TQFTs are described as n-category representation?s
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		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 for $k \ge n+2$
1995		John Baez-James Dolan		Extended TQFT hypothesis?: 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 $n$Hilb.
1995		Valentin Lychagin		Categorical quantization?
1995		Pierre Deligne-Vladimir Drinfel'd-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		Multitope?s, 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. The combination of model category theory and Segal category theory is probably one of the most efficient way of doing simplicial homotopy theory
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 element?s. 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 characteristic?s 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 and it is unlikely that they exist globally
1998		John Baez		Spin foam models?: A 2-dimensional cell complex with faces labeled by representations and edges labeled by intertwining operator?s. 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 scheme?s: 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 $(\vert{n}\vert+\vert{m}\vert)/2$ simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs $\vert{n}\vert$ points on the bottom and $\vert{m}\vert$ points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley-Lieb categories are categorized Temperley-Lieb algebra?s
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		Yakov 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
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 space?s are introduced at the same time
2001		Charles Rezk		Model topos?es and their generalization homotopy topos?es (a model topos without the t-completeness assumption)
2002		Bertrand Toën-Gabriele Vezzosi		Segal topos?es 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 (2/3 volumes published as of 2008)
2002		Dennis Gaitsgory-Kari Vilonen-Edward Frenkel		Proves the geometric Langlands program for GL(n) 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(n) 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 $\lambda$-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 (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 allow to define homotopical versions of initial and terminal objects, limit? and colimit functors (that are computed by local constructions in the book), completeness? and cocompleteness, adjunctions, Kan extensions and universal properties
2004		Dominic Verity		Proves the Street-Roberts conjecture?
2004		Ross Street		Definition of the descent weak ω-category of a cosimplicial weak ω-category
2004		Ross Street		Characterization theorem for cosmoses?: A bicategory M is a cosmos? iff there exists a base bicategory W such that M is biequivalent to ModW. W can be taken to be any full subbicategory of M whose objects form a small Cauchy generator?
2004		Ross Street-Brian Day		Quantum categories? and quantum groupoids?: A quantum category over a braided monoidal category V is an object R with an opmorphism? h:Rop ⊗ R → A into a pseudomonoid A such that h* is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, h and A lie in the autonomous monoidal bicategory Comod(V)co of comonoids. Comod(V)=Mod(Vop)coop. Quantum categories were introduced to generalize Hopf algebroids and groupoids. A quantum groupoid is a Hopf algebra with several objects
2004		Stephen Stolz?-Peter Teichner		Definition of nD QFT of degree p parametrized by a manifold
2004		Stephen 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		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 module?s
2006		Aslak 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 generalize the common concepts of category theory to higher categories and defines n-topose?s, ∞-toposes, sheaves of n-types, ∞-site?s, ∞-Yoneda lemma and proves Lurie characterization theorem? for higher dimensional toposes. Lurie’s theory of higher toposes can be interpreted as giving a good theory of sheaves taking values in ∞-categories. Roughly an ∞-topos is an ∞-category which looks like the ∞-category of all homotopy types. In a topos mathematics can be done. In a higher topos not only mathematics can be done but also “n-geometry”, which is higher homotopy theory?. The topos hypothesis? is that the (n+1)-category nCat is a Grothendieck (n+1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting
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 (or D-modules) on the affine Grassmannian GrG=G((t))/G[[t]] of the original group G
2008		Bruce Bartlett		Primacy of the point hypothesis: An n-dimensional unitary extended TQFT is completely described by the n-Hilbert space it assigns to a point. This is a reformulation of the cobordism hypothesis
2008		Ieke Moerdijk-Clemens Berger		Extends and improved the definition of Reedy category to become invariant under equivalence of categories
2008		Valery Lunts-Dmitri Orlov		The derived categories of coherent sheaves on quasiprojective varieties have unique dg-enhancements
2008		Mike Hopkins-Jacob Lurie		Detailed outline of proof of Baez-Dolan tangle hypothesis and Baez-Dolan cobordism hypothesis which classify extended TQFT in all dimensions

Unclassifiable by year

• EGA (Éléments de géométrie algébrique)
• FGA (Fondements de la Géometrie Algébrique)
• SGA (Séminaire de géométrie algébrique)

Literature

For more on the history of higher category theory, see: * John Baez, Aaron Lauda, A Prehistory of n-Categorical Physics (draft version). * Ross Street, A Conspectus of Australian Category Theory

and for closely related timeline of homological algebra a comprehensive 40-page article by Weibel contains a wealth of insight (and possibly corrections to some things on this page!)

• Charles Weibel, A history of homological algebra

References

$[1]$ On the theory of elimination, Cambridge and Dublin Math. J. 3, 116-120

$[2]$ in his thesis on abelian categories 1962, Bull. Soc. Math. France

$[3]$ “Coherent sheaves on Pn and problems in linear algebra”, Funktsional. Anal. I Prilozhen. 12 (3): 68–69

$[4]$ Cartan Seminaire writing up sheaf theory in 1948 for the first time