nLab Timeline of category theory and related mathematics

This is a timeline of category theory and related mathematics. For discussion about its aims and possible guidelines for people who wish to work on it, see the bottom.

Timeline

YearContributorsEvent
1848Arthur Cayley[1]Computations with complexes, Koszul resolutions, notion of exactness; main result: the resultant is a determinant of a Koszul complex.
1890David Hilbertresolution? and free resolution of modules
1890David HilbertHilbert syzygy theorem?
1893David Hilbertfundamental 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
1894Henri Poincaréfundamental group of a topological space
1895Henri Poincarésimplicial homology
1895Henri Poincaréfundamental work Analysis Situs?, the beginning of algebraic topology
1923Otto Künneth?Künneth formula for (co)homology of product of spaces
1926Otto SchreierClassifies nonabelian extensions of groups having implicitly notions of a pseudofunctor and nonabelian cohomology in dimensions up to 3
1929Walther Mayer?chain complexes
1930Ernst ZermeloAbraham FraenkelStatement of the final ZF-axioms of set theory after being first stated in 1908 and improved upon since then
1932Georges de Rhamde Rham theorem: For a smooth manifold the de Rham cohomology is isomorphic to the singular cohomology with coefficients in R.
1932Eduard ČechČech cohomology, higher homotopy groups of a topological space though nobody paid attention because they were all abelian.
1933Solomon Lefschetzsingular homology of a topological space
1934Reinhold BaerExt groups, Ext functor (for abelian groups and with different notation)
1935Witold Hurewiczhigher homotopy groups of a topological space
1936Marshall StoneStone representation theorem for Boolean algebras initiates various Stone dualities
1937Richard BrauerCecil Nesbitt?Frobenius algebras
1938Hassler Whitney“Modern” definition of cohomology, summarizing the work since James Alexander? and Andrey Kolmogorov? first defined cochains
1940Reinhold Baerinjective modules
1940Kurt Gödel?Paul Bernays?proper classes
1940Heinz HopfHopf algebras
1941Witold Hurewiczfirst 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
1942Samuel EilenbergSaunders Mac Laneuniversal coefficient theorem for Čech cohomology, later this became the general universal coefficient theorem. The notations $\mathrm{Hom}$ and $\mathrm{Ext}$ first appear in their paper
1943Norman Steenrodhomology with local coefficients?
1943Israel GelfandMark Naimark?Gelfand-Naimark theorem (sometimes called Gelfand isomorphism theorem): The category $\mathrm{Haus}$ of locally compact Hausdorff spaces with continuous proper maps as morphisms is equivalent to the category ${C}^{*}\mathrm{Alg}$ of commutative ${C}^{*}$-algebras with proper $*$-homomorphisms as morphisms
1944Garrett Birkhoff?Øystein OreGalois connections generalizing the Galois correspondence: a pair of adjoint functors between two categories that arise from partially ordered sets (in modern formulation)
1944Samuel Eilenberg“Modern” definition of singular homology and singular cohomology
1945Beno EckmannDefines the cohomology ring building on Heinz Hopf's work
1945Saunders Mac LaneSamuel Eilenbergstart of category theory: axioms for categories, functors and natural transformations
1945Norman SteenrodSamuel EilenbergEilenberg-Steenrod axioms for homology and cohomology
1945Jean LerayStarts 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
1945Jean Leraysheaf cohomology
1946Jean Lerayinvents 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
1948Cartan seminar (query 4 down)writes up sheaf theory for the first time
1948A. L. Blakerscrossed 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 ω-groupoids. They form a category $\mathrm{Crs}$ that has many satisfactory properties such as a monoidal structure.
1949André Weilformulates 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
1950Henri Cartanin 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
1950John Henry Whitehead?outlines algebraic homotopy program for describing, understanding and calculating homotopy types of spaces and homotopy classes of mappings
1950Samuel Eilenberg–Joe Zilbersimplicial 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
1951Henri Cartanmodern definition of sheaf theory
1951M M Postnikovpublishes the results of his thesis: Postnikov system
1952William Massey?invents exact couples for calculating spectral sequences
1953Jean-Pierre SerreSerre C-theory? and Serre subcategories
1955Jean-Pierre Serreshows 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)
1955Jean-Pierre Serrecoherent sheaf cohomology? in algebraic geometry
1955Michel LazardIntroduces “analyseurs”, a version of the future operads of Peter May
1956Jean-Pierre SerreGAGA correspondence?
1956Henri CartanSamuel Eilenberginfluential book: Homological Algebra, summarizing the state of the art in its topic at that time. The notation ${\mathrm{Tor}}_{n}$ and ${\mathrm{Ext}}^{n}$, 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
1956Daniel Kansimplicial homotopy theory also called categorical homotopy theory: a homotopy theory completely internal to the category of simplicial sets
1957Charles EhresmannJean Bénaboupointless topology building on Marshall Stone's work
1957Alexander Grothendieckabelian categories in homological algebra that combine exactness and linearity
1957Alexander Grothendieckinfluential 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
1957Alexander Grothendieckthe Grothendieck relative point of view?, S-schemes
1957Alexander GrothendieckGrothendieck-Hirzebruch-Riemann-Roch theorem? for smooth schemes
1957Daniel KanKan 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.
1958Alexander Grothendieckstarts 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
1958Roger 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
1958Daniel Kanlimits in category theory
1958Alexander GrothendieckIntroduces pseudofunctors and descent theory in FGA but publish them later with Pierre Gabriel in SGA1 1961 modernized into fibred categories.
1959Alexander GrothendieckIntroduces formal algebraic geometry and formal schemes (partly with Pierre Cartier) in a seminar Bourbaki and publish it in FGA.
1959Bernard Dworkproves the rationality part of the Weil conjectures (the first conjecture)
1960Alexander Grothendieckfiber functors
1960Daniel KanKan extensions
1960Alexander Grothendieckrepresentable functors
1960Alexander Grothendieckcategorizes Galois theory (Grothendieck Galois theory)
1960Alexander Grothendieckdescent 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
1960Pierre 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.
1961Alexander Grothendiecklocal cohomology?. Introduced at a seminar in 1961 but the notes are published in 1967
1961Jim Stasheffassociahedra later used in the definition of weak n-categories
1961Richard SwanShows 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 $C\left(X\right)$ of continuous functions on $X$ (Serre-Swan theorem)
1963Frank AdamsSaunders Mac LanePROP 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
1963Alexander Grothendiecketale topology, a special Grothendieck topology on schemes
1963Alexander Grothendiecketale cohomology
1963Alexander GrothendieckGrothendieck toposes, which are categories which are like universes (generalized spaces) of sets in which one can do mathematics
1963William Lawverealgebraic theories and algebraic categories
1963William LawvereFounds 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
1963Jean-Louis Verdierafter the advice of Grothendieck, defined triangulated categories and triangulated functors including the main examples: derived categories. Studied derived functors in the triangualted setup
1963Jim Stasheff${A}_{\infty }$-algebras: dg-algebra analogs of topological monoid?s associative up to homotopy appearing in topology (i.e. H-spaces)
1963Jean GiraudGiraud characterization theorem characterizing Grothendieck toposes as categories of sheaves over a small site
1963Charles Ehresmanninternal category theory: internalization of categories in a category $V$ with pullbacks replacing the category $\mathrm{Set}$ (same for classes instead of sets) by $V$ in the definition of a category. Internalization is a way to rise the categorical dimension
1963Charles Ehresmannmultiple categories? and multiple functors?
1963Saunders Mac Lanemonoidal 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
1963Saunders Mac LaneMac Lane coherence theorem? for determining commutativity of diagrams in monoidal categories
1964William LawvereETCS (Elementary Theory of the Category of Sets): An axiomatization of the category of sets which is also the constant case of an elementary topos
1964Barry Mitchell–Peter FreydMitchell-Freyd embedding theorem?: Every small abelian category admits an exact and full embedding into the category of (left) modules ${\mathrm{Mod}}_{R}$ over some ring $R$
1964Rudolf HaagDaniel Kastleralgebraic quantum field theory after ideas of Graeme Segal
1964Alexander Grothendiecktopologizes 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
1964Alexander Grothendieck$l$-adic cohomology?
1964Alexander Grothendieckproves the Weil conjectures except the analogue of the Riemann hypothesis
1964Alexander Grothendiecksix operations formalism in homological algebra; $R{f}_{*}$, ${f}^{-1}$, $R{f}_{!}$, ${f}_{!}$, ${\otimes }^{L}$, $R\mathrm{Hom}$, and proof of its closedness
1964Alexander Grothendieckintroduced 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 $h\left(X\right)$ to each smooth projective variety $X$. When $X$ is not smooth or projective, $h\left(X\right)$ 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 $Q$-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
1965Edgar Brownabstract homotopy categories: a proper framework for the study of the homotopy theory of CW complexes
1965Max Kellydg-categories
1965Max KellySamuel Eilenbergenriched category theory: Categories $C$ enriched over a category $V$ are categories with Hom-sets ${\mathrm{Hom}}_{C}$ 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
1965Charles Ehresmanndefines both strict 2-categories and strict n-categories
1966Alexander Grothendieckcrystals (a kind of sheaf used in crystalline cohomology)
1966William LawvereETAC? (Elementary theory of abstract categories), first proposed axioms for $\mathrm{Cat}$ or category theory using first-order logic
1967Jean Bénaboubicategories (weak $2$-categories) and weak $2$-functors
1967William Lawverefounds synthetic differential geometry
1967Simon KochenErnst SpeckerKochen-Specker theorem in quantum mechanics
1967Jean-Louis Verdierfollwing Grothendieck’s advice, defined triangulated categories and constructed derived categories; redefinition of derived functors in terms of triangulated categories
1967Peter Gabriel–Michel ZismanFamous book “Calculus of fractions and homotopy theory” sets a standard on the categorical approach to localization and axiomatizes simplicial homotopy theory.
1967Daniel QuillenQuillen 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 $hC=C\left[{W}^{-1}\right]$ 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
1967Daniel Quillenhomotopical 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
1967Daniel Quillenfirst fundamental theorem of simplicial homotopy theory?: The category of simplicial sets is a (proper) closed (simplicial) model category
1967Daniel Quillensecond fundamental theorem of simplicial homotopy theory?: The realization functor? and the singular functor? constitute an equivalence of categories $h\Delta$ and $h\mathrm{Top}$ ($\Delta$ the category of simplicial sets)
1967Jean Bénabou$V$-actegories: a category $C$ with an action $\otimes :V×C\to C$ 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 $R$
1968Chen Yang?Rodney Baxter?Yang-Baxter equation, later used as a relation in braided monoidal categories for crossings of braids
1968Alexander Grothendieckcrystalline 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
1968Alexander GrothendieckGrothendieck connection
1968Alexander Grothendieckformulates the standard conjectures on algebraic cycles
1968Michael Artinalgebraic spaces in algebraic geometry as a generalization of schemes
1968Charles Ehresmannsketches: 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
1968Joachim Lambekmulticategories
1969Max KellyNobuo Yoneda?ends and coends
1969Pierre DeligneDavid MumfordDeligne–Mumford stacks as a generalization of schemes
1969William Lawveredoctrines, a doctrine is a monad on a $2$-category
1970William Lawvere-Myles TierneyElementary 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
1970John Conway?Skein theory? of knots: The computation of knot invariants by skein module?s. Skein modules can be based on quantum invariant?s
1970Jean Bénabou-Jacques Roubaudconnect the descent in fibered categories with monadic descent: Benabou-Roubaud theorem
1971Saunders Mac LaneInfluential book: Categories for the working mathematician, which became the standard reference in category theory
1971Horst 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.
1971Harold Temperley?-Elliott Lieb?Temperley-Lieb algebra?s: Algebras of tangle?s defined by generators of tangles and relations among them
1971William Lawvere-Myles TierneyLawvere-Tierney topology on a topos
1971William Lawvere-Myles TierneyTopos theoretic forcing? (forcing in toposes): Categorization of the set theoretic forcing? method to toposes for attempts to prove or disprove the continuum hypothesis, independence of the axiom of choice, etc. in toposes
1971Bob Walters-Ross StreetYoneda structure?s on 2-categories
1971Roger PenroseString diagram?s to manipulate morphisms in a monoidal category
1971Jean Giraud?Gerbe?s: Categorified principal bundles that are also special cases of stacks
1971Joachim LambekGeneralizes the Haskell-Curry-William-Howard correspondence? to a three way isomorphism between types, propositions and objects of a cartesian closed category
1972Max KellyClubs (category theory)? and coherence (category theory)?. A club is a special kind of 2-dimensional theory or a monoid in Cat/(category of finite sets and permutations P), each club giving a 2-monad on Cat
1972John IsbellLocales: 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
1972Peter FreydFundamental 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
1972Alexander GrothendieckUniverses for sets
1972Jean Bénabou-Ross StreetCosmoses 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.
1972Peter MayOperads: 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.
1972William Mitchell-Jean BénabouMitchell-Bénabou language of a topos
1973Chris ReedyReedy 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
1973Kenneth Brown-Stephen GerstenShows 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
1973Kenneth BrownGeneralized 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
1973William LawvereFinds 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
1973Jean BénabouDistributors? (also called modules, profunctors, directed bridges?)
1973Pierre DeligneProves the last of the Weil conjectures, the analogue of the Riemann hypothesis
1973John Boardman-Rainer VogtSegal 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.
1973Daniel QuillenFrobenius 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
1974Michael ArtinGeneralizes Deligne-Mumford stacks? to Artin stacks?
1974Robert ParéParé monadicity theorem?: E is a topos→E° is monadic over E
1974Andy MagidGeneralizes Grothendiecks Galois theory from groups to the case of rings using Galois groupoids
1974Jean BénabouLogic of fibred categories?
1974John Gray?Gray categories with Gray tensor product
1974Kenneth BrownWrites a very influential paper that defines Browns categories? of fibrant objects and dually Brown categories of cofibrant objects
1974Shiing-Shen Chern-James SimonsChern-Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D
1975Saul Kripke?-Andre JoyalKripke-Joyal semantics of the Mitchell-Bénabou internal language? for toposes: The logic in categories of sheaves is first order intuitionistic predicate logic
1975Radu DiaconescuDiaconescu theorem: The internal axiom of choice holds in a topos → the topos is a boolean topos. So in IZF the axiom of choice implies the law of excluded middle
1975Manfred SzaboPolycategories?
1975William LawvereObserves that Delignes theorem? about enough points in a coherent topos implies the Gödel completeness theorem? for first order logic in that topos
1976Alexander GrothendieckSchematic homotopy type?s
1976Marcel CrabbeHeyting 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.
1977Peter JohnstoneVery influential book “Topos theory” (circulated as a preprint a year earlier).
1977Michael Makkai-Gonzalo ReyesDevelops the Mitchell-Bénabou internal language? of a topos thoroughly in a more general setting
1977Andre Boileau-Andre Joyal-Jon ZangwillLST Local set theory?: Local set theory is a typed set theory? whose underlying logic is higher order intuitionistic logic. It is a generalization of classical set theory, in which sets are replaced by terms of certain types. The category C(S) built out of a local theory S whose objects are the local sets (or S-sets) and whose arrows are the local maps (or S-maps) is a linguistic topos?. Every topos E is equivalent to a linguistic topos C(S(E))
1977John 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?
1978John Roberts?Complicial set?s (simplicial sets with structure or enchantment)
1978Francois Bayen-Moshe Flato-Chris Fronsdal-Andre Lichnerowicz-Daniel SternheimerDeformation quantization?, later to be a part of categorical quantization
1978Andre JoyalCombinatorial species? in enumerative combinatorics?
1978Don AndersonBuilding on work of Kenneth Brown defines ABC (co)fibration categories? for doing homotopy theory and more general ABC model categories?, but the theory lies dormant until 2003. Every Quillen model category is an ABC model category. A difference to Quillen model categories is that in ABC model categories fibrations and cofibrations are independent and that for an ABC model category MD is an ABC model category. To 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
1978-1979Alexander 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.
1979Don AndersonAnderson axioms? for homotopy theory in categories with a fraction functor?
1980Alexander Zamolodchikov?Zamolodchikov equation also called tetrahedron equation?
1980Ross StreetBicategorical Yoneda lemma
1980Masaki Kashiwara-Zoghman MebkhoutProves the Riemann-Hilbert correspondence for complex manifolds
1980Peter FreydNumerals? in a topos
1981Shigeru MukaiMukai-Fourier transform?
1982Bob WaltersEnriched categories? with bicategories as a base
1982Martin HylandDevises the effective topos, an environment for recursive mathematics
1983Alexander GrothendieckPursuing stacks: Correspondence by mail with Daniel Quillen about Alexander Grothendieck’s mathematical visions written down in a 629 pages manuscript
1983Alexander GrothendieckFirst appearance of strict ∞-categories in pursuing stacks
1983Alexander GrothendieckFundamental infinity groupoid?: A complete homotopy invariant Π(X) for CW-complexes X. The inverse functor is the geometric realization functor $\mid .\mid$ and together they form an “equivalence” between the category of CW-complexes? and the category of ω-groupoids
1983Alexander GrothendieckHomotopy hypothesis?: The homotopy category of CW-complexes is Quillen equivalent to a homotopy category of reasonable weak ∞-groupoids
1983Alexander GrothendieckGrothendieck derivators: A model for homotopy theory similar to Quilen model categories but more satisfactory. Grothendieck derivators are dual to Heller derivators
1983Alexander GrothendieckElementary 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
1983Alexander GrothendieckSmooth functor?s and proper functors
1984Vladimir Bazhanov-Razumov StroganovBazhanov-Stroganov d-simplex equation? generalizing the Yang-Baxter equation and the Zamolodchikov equation
1984Horst HerrlichUniversal 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
1984Andre JoyalSimplicial sheaves (sheaves with values in simplicial sets). Simplicial sheaves on a topological space X is a model for the hypercomplete? ∞-topos Sh(X)^
1984Andre JoyalShows that the category of simplicial objects in a Grothendieck topos has a closed model structure
1984Andre Joyal-Myles TierneyMain Galois theorem for toposes?: Every topos is equivalent to a category of étale presheaves on an open étale groupoid
1985Michael Schlessinger-Jim StasheffL-algebras
1985Andre Joyal-Ross StreetBraided monoidal categories?
1985Andre Joyal-Ross StreetJoyal-Street coherence theorem? for braided monoidal categories
1985Paul Ghez-Ricardo Lima-John Roberts?C*-categories
1986Joachim Lambek-Phil ScottInfluential book: Introduction to higher order categorical logic
1986Joachim Lambek-Phil ScottFundamental 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
1986Peter Freyd-David YetterConstructs the (compact braided) monoidal category of tangles?
1986Vladimir 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
1986Saunders Mac LaneMathematics, form and function? (a foundation of mathematics)
1987Jean-Yves GirardLinear logic?: The internal logic of a linear category (an enriched category with its Hom-set?s being linear spaces)
1987Peter FreydFreyd representation theorem? for Grothendieck toposes
1987Ross StreetDefinition of the nerve of a weak n-category and thus obtaining the first definition of weak n-category using simplices
1987Ross Street-John Roberts?Formulates Street-Roberts conjecture?: Strict ω-categories are equivalent to complicial sets
1987Andre Joyal-Ross Street-Mei Chee ShumRibbon categories?: A balanced rigid braided monoidal category
1987Iain AitchisonBottom up Pascal triangle algorithm? for computing nonabelian n-cocycle conditions for nonabelian cohomology
1987Vladimir Drinfel'd-Gerard LaumonFormulates geometric Langlands program
1987Vladimir TuraevStarts 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
1988Alex HellerHeller axioms? for homotopy theory as a special abstract hyperfunctor?. A feature of this approach is a very general localization
1988Alex HellerHeller derivator?s, the dual of Grothendieck derivator?s
1988Alex HellerGives 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
1988Graeme SegalElliptic 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
1988Graeme SegalConformal field theory CFT?: A symmetric monoidal functor Z:nCob'''C'''→Hilb satisfying some axioms
1988Edward WittenTopological quantum field theory TQFT: A monoidal functor Z:nCob→Hilb satisfying some axioms
1988Edward WittenTopological string theory?
1989Hans BauesInfluential book: Algebraic homotopy?
1989Michael Makkai-Robert ParéAccessible 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
1989Edward WittenWitten functional integral? formalism and Witten invariant?s for manifolds
1990Alexei Bondal-Mikhail KapranovEnhanced triangulated categories
1990Peter FreydAllegories (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.
1990Nicolai Reshetikhin-Vladimir Turaev-Edward WittenReshetikhin-Turaev-Witten invariant?s of knots from modular tensor categories of representations of quantum groups
1990Cartier et al.Grothendieck Festschrift in 3 volumes with historical contributions including Thomason-Troubaugh article on algebraic K-theory; Categories Tannakiennes? and Breen’s “Bitorseurs et cohomologie nonabeliennes”
1991Jean-Yves GirardPolarization? of linear logic
1991Ross StreetParity complex?es. A parity complex generates a free ω-category
1991Andre Joyal-Ross StreetFormalization 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?
1991Ross StreetDefinition of the descent strict ω-category of a cosimplicial strict ω-category
1991Ross StreetTop down excision of extremals algorithm? for computing nonabelian n-cocycle conditions for nonabelian cohomology
1992Yves Diers?Axiomatic categorical geometry? using algebraic-geometric categories? and algebraic-geometric functors?
1992Saunders Mac Lane-Ieke MoerdijkInfluential book: Sheaves in geometry and logic
1992John Greenlees-Peter MayGreenlees-May duality?
1992Vladimir TuraevModular 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?
1992Vladimir Turaev-Oleg Viro?Turaev-Viro state sum model?s based on spherical categories (the first state sum models) and Turaev-Viro state sum invariant?s for 3-manifolds
1993Paul TaylorASD (Abstract Stone Duality): A reaxiomatisation of the notions of space and map in general topology in terms of λ-calculus of computable continuous functions and predicates that is both constructive and computable
1993Ruth LawrenceExtended TQFTs
1993David Yetter-Louis CraneCrane-Yetter state sum model?s based on ribbon categories and Crane-Yetter state sum invariant?s for 4-manifolds
1993Kenji FukayaA-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.
1993John Barrett?-Bruce WestburySpherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane
1993Maxim KontsevichKontsevich 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
1993Daniel FreedA new view on TQFT using modular tensor categories that unifies 3 approaches to TQFT (modular tensor categories from path integrals)
1994Francis BorceuxHandbook of categorical algebra (3 volumes)
1994Jean Bénabou-Bruno LoiseauOrbitals? in a topos
1994Maxim KontsevichFormulates 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)
1994Louis Crane-Igor FrenkelHopf categories and construction of 4D TQFTs by them
1994John FischerDefines the 2-category of 2-knot?s (knotted surfaces)
1995Bob Gordon-John Power-Ross StreetTricategories and a corresponding coherence theorem: Every weak 3-category is equivalent to a Gray 3-category which is a much simpler
1995Ross Street-Dominic VeritySurface diagram?s for tricategories
1995Sjoerd CransA general procedure of transferring closed model structures on a category along adjoint functor pairs to another category
1995André Joyal-Ieke MoerdijkAST 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?
1995Michael MakkaiSFAM 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
1995John Baez-James DolanOpetopic set?s (opetopes) based on operads. Weak n-categories are n-opetopic sets
1995John Baez-James DolanIntroduces the periodic table of k-tuply monoidal n-categories.
1995John Baez-James DolanOutlines a program in which n-dimensional TQFTs are described as n-category representation?s
1995John Baez-James Dolantangle hypothesis: The $n$-category of framed $n$-tangles in $n+k$ dimensions is $\left(n+k\right)$-equivalent to the free weak $k$-tuply monoidal $n$-category with duals on one object
1995John Baez-James DolanStabilization 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 $k\ge n+2$
1995John Baez-James DolanExtended 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.
1995Valentin LychaginCategorical quantization?
1995Pierre Deligne-Vladimir Drinfel'd-Maxim KontsevichDerived 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
1997Maxim KontsevichFormal deformation quantization theorem: Every Poisson manifold admits a differentiable star product? and they are classified up to equivalence by formal deformations of the Poisson structure
1998Claudio Hermida-Michael-Makkai?-John PowerMultitope?s, Multitopic sets
1998Carlos SimpsonSimpson conjecture: Every weak ∞-category is equivalent to a ∞-category in which composition and exchange laws are strict and only the unit laws are allowed to hold weakly. It is proven for 1,2,3-categories with a single object
1998André Hirschowitz-Carlos SimpsonGive a model category structure on the category of Segal categories. Segal categories are the fibrant-cofibrant objects and Segal maps are the weak equivalences. In fact they generalize the definition to that of a Segal n-category and give a model structure for Segal n-categories for any n≥1. The combination of model category theory and Segal category theory is probably one of the most efficient way of doing simplicial homotopy theory
1998Chris Isham-Jeremy ButterfieldKochen-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
1998Richard ThomasRichard 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 invariant?s 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
1998John BaezSpin 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
1998John Baez-James DolanMicrocosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure
1998Alexander 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.
1998Maxim KontsevichCalabi-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
1999Joseph Bernstein-Igor Frenkel-Mikhail KhovanovTemperley-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 $\left(\mid n\mid +\mid m\mid \right)/2$ simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs $\mid n\mid$ points on the bottom and $\mid m\mid$ points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley-Lieb categories are categorized Temperley-Lieb algebra?s
1999Moira Chas-Dennis SullivanConstructs String topology? by cohomology. This is string theory on general topological manifolds
1999Mikhail KhovanovKhovanov homology: A homology theory for knots such that the dimensions of the homology groups are the coefficients of the Jones polynomial of the knot
1999Vladimir TuraevHomotopy quantum field theory HQFT
1999Vladimir Voevodsky-Fabien MorelConstructs the homotopy category of schemes?
1999Ronald Brown?-George Janelidze2-dimensional Galois theory
2000Vladimir VoevodskyGives two constructions of motivic cohomology of varieties, by model categories in homotopy theory and by a triangulated category of DM-motives
2000Yasha 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
2001Charles RezkConstructs a model category with certain generalized Segal categories as the fibrant objects, thus obtaining a model for a homotopy theory of homotopy theories. Complete Segal space?s are introduced at the same time
2001Charles RezkModel topos?es and their generalization homotopy topos?es (a model topos without the t-completeness assumption)
2002Bertrand Toën-Gabriele VezzosiSegal topos?es coming from Segal topologies?, Segal sites? and stacks over them
2002Bertrand Toën-Gabriele VezzosiHomotopical 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)
2002Peter 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)
2002Dennis Gaitsgory-Kari Vilonen-Edward FrenkelProves the geometric Langlands program for GL(n) over finite fields
2003Denis-Charles CisinskiMakes further work on ABC model categories? and brings them back into light. From then they are called ABC model categories after their contributors
2004Dennis GaitsgoryExtended 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
2004Mario CaccamoFormal category theoretical expanded λ-calculus? for categories
2004Francis Borceux-Dominique BournHomological categories?
2004William Dwyer-Philips Hirschhorn-Daniel Kan-Jeffrey SmithIntroduces in the book: Homotopy limit functors on model categories and homotopical categories, a formalism of homotopical categories and homotopical functors (weak equivalence preserving functors) that generalize the model category formalism of Daniel Quillen. A homotopical category has only a distinguished class of morphisms (containing all isomorphisms) called weak equivalences and satisfy the two out of six axiom. This 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
2004Dominic VerityProves the Street-Roberts conjecture?
2004Ross StreetDefinition of the descent weak ω-category of a cosimplicial weak ω-category
2004Ross StreetCharacterization 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?
2004Ross Street-Brian DayQuantum 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
2004Stephen Stolz?-Peter TeichnerDefinition of nD QFT of degree p parametrized by a manifold
2004Stephen Stolz?-Peter TeichnerGraeme Segal proposed in the 1980s to provide a geometric construction of elliptic cohomology (the precursor to tmf?) as some kind of moduli space of CFTs. Stephan Stolz and Peter Teichner continued and expanded these ideas in a program to construct TMF? as a moduli space of supersymmetric Euclidean field theories. They conjectured a Stolz-Teichner picture? (analogy) between classifying spaces of cohomology theories in the chromatic filtration (de Rham cohomology,K-theory,Morava K-theories) and moduli spaces of supersymmetric QFTs parametrized by a manifold (proved in 0D and 1D)
2005Peter SelingerDagger categories? and dagger functor?s. Dagger categories seem to be part of a larger framework involving n-categories with duals?
2005Peter Ozsváth?-Zoltán Szabó?Knot Floer homology?
2006P. Carrasco-A.R. Garzon-E.M. VitaleCategorical crossed module?s
2006Aslak Buan-Robert Marsh-Markus Reineke-Idun Reiten-Gordana TodorovCluster categories?: Cluster categories are a special case of triangulated Calabi-Yau categories of Calabi-Yau dimension 2 and a generalization of cluster algebras
2006Jacob LurieMonumental 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
2007Bernhard Keller-Thomas Hughd-cluster categories?
2007Dennis Gaitsgory-Jacob LuriePresents 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
2008Bruce BartlettPrimacy 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
2008Ieke Moerdijk-Clemens BergerExtends and improved the definition of Reedy category to become invariant under equivalence of categories
2008Valery Lunts-Dmitri OrlovThe derived categories of coherent sheaves on quasiprojective varieties have unique dg-enhancements
2008Mike Hopkins-Jacob LurieDetailed 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:

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!)

References

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

Rafael: Zoran Škoda, you have to be more precise. Koszul resolutions of what? Notion of exactness for what?

Zoran: he does one VERY complex calculation in which he masterly uses constructions involving organizational principles as above, without modern words and general definitions. The article is reproduced in the book Gel’fand, Kapranov, Zelevinsky which is about the current state of the art in similar computations and which can explain more.

Rafael Borowiecki: Me and Zoran Škoda are arguing if to keep this first entry in the timeline. See 2009 September changes 2009-09-04 for the start of the discussion (that could be moved here). I am waiting to hear what others think about it.

Rafael: After thinking a bit more about it, it comes down to the same as i wrote above. Koszul resolution of whitch algebra? Is it a particular kind of Koszul algebra for starter? Otherwise the explanation is no real main theorem of the paper. It is not closed. I suppose the exactness would be for the Koszul resolution.

Then, this theorem is by far the most specialized result in the timeline. So i see it as the least important result in the timeline. If it would at least be a theory of determinants of complexes or inspired this theory, it would be worth mentioning, but it’s not. There are too many particular cases and results to start to include them, the idea was at least to include the most important events in category theory and the related disciplines.

Zoran: I am not talking Koszul algebras of Priddy but the older Koszul resolutions in the subject of commutative algebra. Of course, Koszul pair of the symmetric vs. exterior algebra is behind the classical Koszul resolutions; but the resolutions are typically, like in Cayley’s work, resolutions of modules of finite type over reduced f.g. commutative algebras. Thus take any projective variety and a coherent sheaf on it and you are in the setup of this paper. The paper starts with a notion of resolution in about this setup, then develops Koszul resolutions and calculations with determinants which express the bases of the modules which appear in the resolution. This is very general. Gel’fand, Kapranov, Zelevinsky in their book dedicate chapter 2 (and 3.4) to “Cayley method” and say this about the paper: “The method goes back to the remarkable paper by Cayley Ca4 on elimination theory, in which the foundations were laid for what is now called homological algebra.” But what happened with your too shy question/suggestion to include also Galois ? By the way what does it mean “categorical quantization” in 1978 entry ? Never heard.

Rafael: I am thinking how to include what you wrote above in the Cayley entry. But not right now. Perhaps you can do it better. I am thinking more about if it is right to call it a foundation of homological algebra. In this way one could say that Gauss by Gauss elimination (i think it is the first general method) founded higher dimensional linear algebra. Hmm. I would say that more is needed for a foundation than a calculation. Cayley is doing homological algebra by a general calculation and not developing a new theory. A theory have several interacting definitions and theorems and so on. But i must think more about it.

Categorical quantization means the 1995 (unfortunately empty) Valentin Lychagin entry in the timelie. The idea is that quantization is a functor from the category of integrable Poisson manifolds and symplectomorphisms to the category of separable C${}^{*}$-algebras and $*$-homomorphisms. The object function of the functor is deformation quantization and the morphism function of the functor is geometric quantization. I don’t remember the reference but i think it is one of his papers here.

Well, you did have better motivation than me to include Galois theory. If i write it, it will be one short sentence since i am writing other entries now, so i suggest you write it to show some of its categorical flavour, but not as incomplete as the Cayley entry. I mean leaving questions begging to be answered in the entry.

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

Zoran: No, thesis is 1960. The reprint as a paper is 1962 about which we do have a separate nlab entry which is quoted above with the link hence no need for imprecise footnote heee; Verdier’s thesis on triangulated categories has been published in a journal about 30 years after the fact.

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

Rafael: Zoran Škoda, Which is the other paper? You have to be more precise with the names. Joseph Bernstein? Israel Gel’fand? Gel’fand who? Lets stick to one year per event.

Zoran: No, Beilinson’s work has been published in two papers in separate years; one of the two is written as an accompanying paper to BGG paper (BGG is the abbreviation to search for, BGG resolutions etc; B is Joseph Bernstein of course, G and G are both Gelfands, father and son, Israel Moiseevich and Sergej Israelovich). The references and the work of Beilinson cited and build upon in reference cited in nlab link within above 1990 entry on Bondal-Kapranov. I think that 1980 is too late for attribution to Mebkhout’s thesis, which is earlier than Kashiwara’s work by 1-2 years. Voevodsky’s work is largely built in mid 1990-s (with ideas of his approach starting in late 1980s). 1995 attributes the notion of categorification to Crane; Toen gave a talk that Ehresmann invented and systematically used the notion, but I am not sure if he used the WORD or only the philosophy of it, somebody should check that with French school. One of the major missing points of the list above is that seminal works of Duskin are completely not mentioned, including his pioneering works on higher groupoids and higher torsors in simplicial language; similarly contributions of say Marta Bunge, Dedecker, Getzler… But I am not enthusiastic about pinning history to something as untrue as trying to fit into single years, and would much rather instead more work on other nlab entries.

Toby: What is the purpose of the timeline? If it's to know when things were done, not so much when they were published, then we should generally put entries in the earliest appropriate date; I think that this is what John would like to get out of this article. But if we want to note the publishing date of every paper, then many ideas will be spread out over many dates. Even if we want to know when ideas came, there may still be a range of dates from first glimmer to final working out.

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

Zoran: this is CartanSem 1948-1949. I see no sheaf theory explicitly introduced.

Rafael: I don’t see anything there, the page is missing. In this case i have a reference i trusted. Where this author have his reference i have no idea.

Zoran: oops one character too long, try again

Rafael: Alas, don’t forget i don’t know french. When i read the entry i interpret it as it was written on a blackboard. But something as important as this should be in the proceedings. I also know that some concepts such as the Schwartzian derivative are well hidden in their first appearance. I am open to remove entries, so if you know you are right remove it.

Zoran: In fact I believe you are right, that Cartan seminars had sheaves in 1948; thsi is what I recall from my earlier readings, but it is not in the writeup of the seminars as it stands; I do not know what is the reference. Maybe we should look in Serre’s FAC paper where he should have quoted earlier papers on sheaf theory.

Zoran: Here is the key! The 1948/9 acad year volume of Cartan seminar has 11 exposes, as on numdam. Weibel in his historical survey says that exposes 12-17 in mimeographed version had sheaf theory and that those exposes were abandoned, hence not published for real later and that Cartan again in 1950/51 a reworked version is published as exposes 14-20 (the latter are at numdam). Now timeline has 3 years as 3 contributions of Cartan 1948, 1950, 1951; at least the latter two for one academic year of the same volume of publications is not sensitive.

Rafael: Yes, this is something to include :) Good detective work. Here is another. Is it necessary that the variety is Noetherian in the Serre-Swan theorem? It don’t say so for instance here

Zoran In Serre’s paper, says and uses noetherian. In a manuscript book on alg K-theory, Weibel quotes several related conditions, I think one detail is even there not precise, we had short nonconclusive correspondence on that. I should remember and look in my 2 years or so old letters on that. Another issue is 1964 item on sites. As there is a reference (I have a copy) M, Artin, Grothendieck topologies (mimeographed notes), Harvard University, Cambridge, Mass., 1962. than the year should be before 1962. One should compare descent chapters in FGA which are earlier; Grothendieck created descent theory to do it for qcoh sheaves in faithfully flat Grothendieck topology, so my suspicion would be even as early as 1959, but I am not sure; this is just a response without looking.

Rafael: After much work i found two references. Hazevinkel which gives 1963. And Planetmath suggesting SGA4. Wikipedia gives SGA4 is from 1963-1964 which fits. Even if i am unsure if these are the reference i used in the timeline they all fit.

Discussion

Concerning an update from Wikipedia

Rafael: I have been writing on a new version of this timeline in wikipedia and i can handle the migration to nLab now. Except maby for some formulas. But how do i handle to update most of 1500 links!?

Concerning the Desire for References

Rafael: John, if you need references a priority is needed for which references to include first.

John: I would really like to know who first defined (strict) 2-categories, and in precisely what book or paper! I have a fairly large number of references already, here and in some other expository papers. If I had more time I would transfer a lot of these references to the $n$Lab — though I see on the $n$Forum that Andrew Stacey wants to improve our facilities for dealing with references, and maybe I should wait for that.

In general, people will consult this timeline if they want to know when something was done, but such people often also want to know where it was done. And, no scholar worth his salt would accept a date without any way to check whether it’s correct.

Rafael: The first reference i found was your “An introduction to n-categories” that states that strict 2-categories where first defined by C. Ehresmann in 1965 and by 1962 (page 12). I must have reasoned that since you mention two years and also speak about strict n-categories that were defined in 1965 you must mean that strict 2-categories were defined in 1962, and adopted it in the timeline.

A further search found “A 2-categories companion” - Stephen Lack, where he states that C. Ehresmann defined strict 2-categories in reference 13 which is the 1965 “Categories et structures”. I have changed the year in the timeline.

Then i think you are missing one of the big points with the timeline: to find things (mostly explanations) that are hard to find elsewhere. This might not seem hard to experts in category theory but for the rest “nonexperts” it is. I know how much time i have spend on it. For instance first i could not find any definition of a ${A}_{\infty }$-category for a long time. Then i found 9 definitions (excluding nLab)! Only 2 of them completely agreed! I had to actually make a table with all the differences to compare them. I think i got it right.

Concerning the Desire for ‘Milestones’

Urs: I am very fond of the aim of this entry. I suggest, though (and might start implementing that when I find time), that we try to split, for the reader’s convenience, a list of true generally accepted milestone constributions that indicate the big stepts forward, from a a list that rather aims to give a somewhat complete list of all contributions of some relevance.

Rafael: Why not splitting? The table will grow larger and a split will be necessary soon anyway. I just hope everyone agrees what the milestones are.

John We won’t agree on the milestones, but we can have fun arguing about it. I think this page here should include lots of developments, with just a little text about each one, so people can find out when things happened. Another entry, called something like “Milestones in Category Theory and Related Mathematics”, could be more selective. We should not remove information from this big list, but we can copy “milestones” to a shorter list.

When writing papers it’s often nice to refer to the article or book where a concept was first developed. So, I want to have this information in the $n$Lab eventually. In particular, if we claim that multiple categories were invented in 1963 by Ehresmann, we should have a reference somewhere to back up this claim. But maybe that reference should appear in the (not-yet-existent) page on multiple categories. That will prevent this page from becoming unnecessarily long.

David: I can see the milestone list being contentious. Anyway far more important to improve this timeline first. We could do with some experts looking it over. I’m sure most would include the effective topos concept much sooner than the quantum topos. I’ve transported out some of the material for a couple of lengthy entries, e.g., AST. It looks better to have entries of comparable length and with no breaks in the table, e.g., to define cosmoses.

John: I’ve deleted the entry on ‘quantum topos’, since in fact most experts are very doubtful that a correct definition has been found for this notion, and they’re even uncertain such a notion deserves to exist. Let’s stick to concepts that are fairly widely considered to be fruitful.

Rafael: I don’t mind deleting it. But i am also curious what is wrong with the definition

Urs: that sounds good. Maybe I was thinking that “timeline of category theory” sounds like it shouldn’t contain just every little bit. So maybe if and when we split, is there any majority for keeping this title here for the short list and another title for the long list, maybe literature list of category theory or the like?

John: Right now I don’t feel this timeline contains too much minor stuff. It mostly just contains stuff that’s very important to the history of categories and related math. I might take out a few things, but mostly I’d feel the need to add things to get a reasonably full timeline.

I might shorten some of the entries though: the longer ones seem to be on topics that Rafael knows more about, so their length doesn’t seem proportional to their importance. I also don’t think there need to be 5 entries devoted to my first paper with Jim Dolan!

Someday someone (maybe Andrew Stacey) will create an easy way for us to dump bibliographical information into the $n$Lab, and then maybe we can grow an enormous ‘literature list’ without doing too much work.

Rafael: I actually had the idea to collect John’s first paper in a separate entry. If i don’t find a way to do line breaks a separate page is needed for it. The same goes for pursuing stacks! I am very interesting to hear what you have in mind to add! I would edit more (especially clean up links and style in the timeline) but nLab is too slow for me to edit efficiently.

Toby: See Bibliography for the beginnings of a literature list.

Jim Stasheff: Is bibtex incompatible with this wiki? The nice lines separating entries on screen do not print. It would be better (but too late?) if the person’s name was aligned with the first line of the Event.

Toby: Yet another reason for formatting this all as a list, I think.

Todd Trimble: Can I ask what organizational principles people are using to compile this list? My impression is that huge swaths are being left out (for example, last time I looked I didn’t see where monads/triples were introduced, and I didn’t see the 1965 Eilenberg-Moore paper).

I like the idea of a page with a shorter list of major themes in category theory, each of which could have its own timeline page, rather than some huge monolithic entry on a subject so huge and multifarious as category theory.

Rafael Borowiecki: It is me who is writing this timeline here and at wikipedia, but i like when other people improve it. I have no real principles except to only include the most important events. The wikipedia timeline have at the beginning an explanation for “related”, this is a kind of organizational principle. The wikipedia timeline is much bigger (and contain nLabs version) and i plane to move it to nLab, but you won’t be able to see it until four days or so. It might include what you are missing. I also have a list of things to include. Monads are included here in 1958. The 1965 Eilenberg-Moore paper i don’t know what it is but i might recognize it when i see/hear about it. Please explain. As well as more that you see is missing.

The size of the timeline is a difficult issue since it is a matter of taste. But lets compare advantages vs. disadvantages. I say it is easier to load and search in one page. But the real problem is to classify the entries. Wikipedia gives a try but there will be entries that are outside it or difficult to classify such as Pursuing stacks, is it category theory or topology or algebraic geometry? An advantage of the split is a classification and hence that the reader can choose which subject he wants to read. Anyway, if the split could be done i still would prefer to have the subtimelines on one page. So far at least, maby i change my mind when each of them is as long as this one here at nLab.

Todd Trimble: Thanks, Rafael. It’s clear that you’ve put a great deal of work into this and the one over at wikipedia, and I’m sure this work has been beneficial to you personally. I’ll see what I can do to help.

I wouldn’t consider myself a connoisseur of timelines, but the main benefit I would hope to draw from one is a coherent sense of a particular historical development; it seems to me it would be hard for anyone to attain that coherent sense from a timeline of such an intricately branched field as category theory, no matter how skillfully it was compiled. There are just way too many threads going on – too many dimensions – to mesh well with the single dimension of time. That of course is why I thought breaking it down into major branches and subbranches would be more useful for most readers, although it may be too late for that now.

You’re absolutely right – I overlooked the 1958 entry on Godement’s ‘standard constructions’. The 1965 paper by Eilenberg and (J.C.) Moore is Adjoint functors and triples, Illinois J. Math. 9, 381-395. It gives the very important category of algebras construction (aka Eilenberg-Moore construction) among other things, which led to other studies on monadicity, including for example the Beck conditions and Manes’ theorem (1968 or 1969) that compact Hausdorff spaces are monadic over sets. The tie-in between algebraic theories and monads was one of the major developments of the 1960’s (as you point out in the 1958 entry, but those points came into fruition after 1958).

One theme I see little mention of is coherence theorems and their interaction with categorical proof theory – starting with Lambek’s groundbreaking rendition of Gentzen cut elimination in categorical terms. So far I see only the 1963 entry on Mac Lane’s Rice Studies paper. (Incidentally, I have a lot of trouble believing that credit is due to Lambek and P.J. Scott for observing that ‘fundamental theorem of topology’ listed under 1986.)

Revised on February 20, 2013 14:14:10 by Tim Porter (95.147.236.41)