1971–1980
| Year | Contributors | Event |
|---|---|---|
| 1971 | Saunders Mac Lane | Influential book: Categories for the working mathematician, which became the standard reference in category theory |
| 1971 | Horst Herrlich-Oswald Wyler | Categorical topology: The study of topological categories of structured sets (generalizations of topological spaces, uniform spaces and the various other spaces in topology) and relations between them, culminating in universal topology. General categorical topology study and uses structured sets in a topological category as general topology study and uses topological spaces. Algebraic categorical topology tries to apply the machinery of algebraic topology for topological spaces to structured sets in a topological category. |
| 1971 | Harold Temperley-Elliott Lieb | Temperley–Lieb algebras: Algebras of tangles defined by generators of tangles and relations among them |
| 1971 | William Lawvere–Myles Tierney | Lawvere–Tierney topology on a topos |
| 1971 | William Lawvere–Myles Tierney | Topos theoretic forcing (forcing in toposes): Categorization of the set theoretic forcing method to toposes for attempts to prove or disprove the continuum hypothesis, independence of the axiom of choice, etc. in toposes |
| 1971 | Bob Walters-Ross Street | Yoneda structures on 2-categories |
| 1971 | Roger Penrose | String diagrams to manipulate morphisms in a monoidal category |
| 1971 | Jean Giraud | Gerbes: Categorified principal bundles that are also special cases of stacks |
| 1971 | Joachim Lambek | Generalizes the Haskell-Curry-William-Howard correspondence to a three way isomorphism between types, propositions and objects of a cartesian closed category |
| 1972 | Max Kelly | Clubs (category theory) and coherence (category theory). A club is a special kind of 2-dimensional theory or a monoid in Cat/(category of finite sets and permutations P), each club giving a 2-monad on Cat |
| 1972 | John Isbell | Locales: A "generalized topological space" or "pointless spaces" defined by a lattice (a complete Heyting algebra also called a Brouwer lattice) just as for a topological space the open subsets form a lattice. If the lattice possess enough points it is a topological space. Locales are the main objects of pointless topology, the dual objects being frames. Both locales and frames form categories that are each 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 (mathematics) for sets |
| 1972 | Jean Bénabou–Ross Street | Cosmoses (category theory) which categorize universes: A cosmos is a generalized universe of 1-categories in which you can do category theory. When set theory is generalized to the study of a Grothendieck topos, the analogous generalization of category theory is the study of a cosmos.
Ross Street definition: A bicategory such that Jean Bénabou definition: A bicomplete symmetric monoidal closed category |
| 1972 | Peter May | Operads: An abstraction of the family of composable functions of several variables together with an action of permutation of variables. Operads can be seen as algebraic theories and algebras over operads are then models of the theories. Each operad gives a monad on Top. Multicategories with one object are operads. PROPs generalize operads to admit operations with several inputs and several outputs. Operads are used in defining opetopes, higher category theory, homotopy theory, homological algebra, algebraic geometry, string theory and many other areas. |
| 1972 | William Mitchell-Jean Bénabou | Mitchell-Bénabou internal language of a toposes: For a topos E with subobject classifier object Ω a language (or type theory) L(E) where: 1) the types are the objects of E 2) terms of type X in the variables xi of type Xi are polynomial expressions φ(x1,...,xm):1→X in the arrows xi:1→Xi in E 3) formulas are terms of type Ω (arrows from types to Ω) 4) connectives are induced from the internal Heyting algebra structure of Ω 5) quantifiers bounded by types and applied to formulas are also treated 6) for each type X there are also two binary relations =X (defined applying the diagonal map to the product term of the arguments) and ∈X (defined applying the evaluation map to the product of the term and the power term of the arguments). A formula is true if the arrow which interprets it factor through the arrow true:1→Ω. The Mitchell-Bénabou internal language is a powerful way to describe various objects in a topos as if they were sets and hence is a way of making the topos into a generalized set theory, to write and prove statements in a topos using first order intuitionistic predicate logic, to consider toposes as type theories and to express properties of a topos. Any language L also generates a linguistic topos E(L) |
| 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 assumptions on the topological space |
| 1973 | Kenneth Brown | Generalized sheaf cohomology of a topological space X with coefficients a sheaf on X with values in Kans category of spectra with some finiteness conditions. It generalizes generalized cohomology theory and sheaf cohomology with coefficients in a complex of abelian sheaves |
| 1973 | William Lawvere | Finds that Cauchy completeness can be expressed for general enriched categories with the category of generalized metric spaces as a special case. Cauchy sequences become left adjoint modules and convergence become representability |
| 1973 | Jean Bénabou | Distributors (also called modules, profunctors, directed bridges) |
| 1973 | Pierre Deligne | Proves the last of the Weil conjectures, the analogue of the Riemann hypothesis |
| 1973 | 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. Def: A simplicial space X such that X0 (the set of points) is a discrete simplicial set and the Segal map Segal categories are a weak form of S-categories, in which composition is only defined up to a coherent system of equivalences. |
| 1973 | Daniel Quillen | Frobenius categories: An exact category in which the classes of injective and projective objects coincide and for all objects x in the category there is a deflation P(x)→x (the projective cover of x) and an inflation x→I(x) (the injective hull of x) such that both P(x) and I(x) are in the category of pro/injective objects. A Frobenius category E is an example of a model category and the quotient E/P (P is the class of projective/injective objects) is its homotopy category hE |
| 1974 | Michael Artin | Generalizes Deligne–Mumford stacks to Artin stacks |
| 1974 | Robert Paré | Paré monadicity theorem: E is a topos→E° is monadic over E |
| 1974 | Andy Magid | Generalizes 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–André Joyal | Kripke–Joyal semantics of the Mitchell–Bénabou internal language for toposes: The logic in categories of sheaves is first order intuitionistic predicate logic |
| 1975 | Radu Diaconescu | Diaconescu theorem: The internal axiom of choice holds in a topos → the topos is a boolean topos. So in IZF the axiom of choice implies the law of excluded middle |
| 1975 | Manfred Szabo | Polycategories |
| 1975 | William Lawvere | Observes that 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 types |
| 1976 | Marcel Crabbe | Heyting categories also called logoses: Regular categories in which the subobjects of an object form a lattice, and in which each inverse image map has a right adjoint. More precisely a coherent category C such that for all morphisms f:A→B in C the functor f*:SubC(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 | Michael Makkai–Gonzalo Reyes | Develops the Mitchell–Bénabou internal language of a topos thoroughly in a more general setting |
| 1977 | Andre Boileau–André 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 ω-categories. There are two methods of constructing general nonabelian cohomology, as nonabelian sheaf cohomology in terms of descent for ω-category valued sheaves, and in terms of homotopical cohomology theory which realizes the cocycles. The two approaches are related by codescent |
| 1978 | John Roberts | Complicial sets (simplicial sets with structure or enchantment) |
| 1978 | Francois Bayen–Moshe Flato–Chris Fronsdal–Andre Lichnerowicz–Daniel Sternheimer | Deformation quantization, later to be a part of categorical quantization |
| 1978 | André Joyal | Combinatorial species in enumerative combinatorics |
| 1978 | Don Anderson | Building on work of Kenneth Brown defines ABC (co)fibration categories for doing homotopy theory and more general ABC model categories, but the theory lies dormant until 2003. Every Quillen model category is an ABC model category. A difference to Quillen model categories is that in ABC model categories fibrations and cofibrations are independent and that for an ABC model category 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 |
| 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 |
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