Brouwer develops intuitionism as a contribution to foundational debate in the period roughly 1910 to 1930 on mathematics, with intuitionistic logic a by-product of an increasingly sterile discussion on formalism.
Module theory is developed by Noether and her students, and algebraic topology starts to be properly founded in abstract algebra rather than by ad hoc arguments.
Starts sheaf theory: At this time a sheaf was a map that assigned a module or a ring to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its p-th cohomology group.
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.
Crossed complexes (called group systems by Blakers), after a suggestion of Samuel Eilenberg: A nonabelian generalization of chain complexes of abelian groups which are equivalent to strict ω-groupoids. They form a category Crs that has many satisfactory properties such as a monoidal structure.
Formulates the Weil conjectures on remarkable relations between the cohomological structure of algebraic varieties over C and the diophantine structure of algebraic varieties over finite fields.
In the book Sheaf theory from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology with support in an axiomatic form and more.
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.
Modern definition of sheaf theory in which a sheaf is defined using open subsets instead of closed subsets of a topological space and all the open subsets are treated at once. A sheaf on a topological space X becomes a functor resembling a function defined locally on X, and taking values in sets, abelian groups, commutative rings, modules or generally in any category C. In fact Alexander Grothendieck later made a dictionary between sheaves and functions. Another interpretation of sheaves is as continuously varying sets (a generalization of abstract sets). Its purpose is to provide a unified approach to connect local and global properties of topological spaces and to classify the obstructions for passing from local objects to global objects on a topological space by pasting together the local pieces. The C-valued sheaves on a topological space and their homomorphisms form a category.
Yoneda publishes his famous lemma. Yoneda's Lemma allows one to consider objects in a (small) category as a presheaves. Yoneda lemma plays a critical role in the study of representable functors in algebraic geometry. For example, even though it is never mentioned explicitly, it is central to the ideas of Grothendieck's "Fondements de la Géométrie Algébrique".
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.
Starts new foundation of algebraic geometry by generalizing varieties and other spaces in algebraic geometry to scheme which have the structure of a category with open subsets as objects and restrictions as morphisms. form a category that is a Grothendieck topos, and to a scheme and even a stack one may associate a Zariski topos, an étale topos, a fppf topos, a fpqc topos, a Nisnevich topos, a flat topos, ... depending on the topology imposed on the scheme. The whole of algebraic geometry was categorized with time.
Monads in category theory (then called standard constructions and triples). Monads generalize classical notions from universal algebra and can in this sense be thought of as an algebraic theory over a category: the theory of the category of T-algebras. An algebra for a monad subsumes and generalizes the notion of a model for an algebraic theory.
Descent theory: An idea extending the notion of gluing in topology to scheme to get around the brute equivalence relations. It also generalizes localization in topology
Shows there is a 1−1 correspondence between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C(X) of continuous functions on X (Serre–Swan theorem)
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
Founds categorical logic, discovers internal logics of categories and recognizes its importance and introduces Lawvere theories. Essentially categorical logic is a lift of different logics to being internal logics of categories. Each kind of category with extra structure corresponds to a system of logic with its own inference rules. A Lawvere theory is an algebraic theory as a category with finite products and possessing a "generic algebra" (a generic group). The structures described by a Lawvere theory are models of the Lawvere theory
Internal category theory: Internalization of categories in a category V with pullbacks is replacing the category Set (same for classes instead of sets) by V in the definition of a category. Internalization is a way to rise the categorical dimension
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
Introduced in a letter to Jean-Pierre Serre conjectural motives to express the idea that there is a single universal cohomology theory underlying the various cohomology theories for algebraic varieties. According to Grothendieck's philosophy there should be a universal cohomology functor attaching a pure motive h(X) to each smooth projective varietyX. When X is not smooth or projective h(X) must be replaced by a more general mixed motive which has a weight filtration whose quotients are pure motives. The category of motives (the categorical framework for the universal cohomology theory) may be used as an abstract substitute for singular cohomology (and rational cohomology) to compare, relate and unite "motivated" properties and parallel phenomena of the various cohomology theories and to detect topological structure of algebraic varieties. The categories of pure motives and of mixed motives are abelian tensor categories and the category of pure motives is also a Tannakian category. Categories of motives are made by replacing the category of varieties by a category with the same objects but whose morphisms are correspondences, modulo a suitable equivalence relation; different equivalences give different theories. Rational equivalence gives the category of Chow motives with Chow groups as morphisms which are in some sense universal. Every geometric cohomology theory is a functor on the category of motives. Each induced functor ρ:motives modulo numerical equivalence→graded Q-vector spaces is called a realization of the category of motives, the inverse functors are called improvements. Mixed motives explain phenomena in as diverse areas as: Hodge theory, algebraic K-theory, polylogarithms, regulator maps, automorphic forms, L-functions, ℓ-adic representations, trigonometric sums, homotopy of algebraic varieties, algebraic cycles, moduli spaces and thus has the potential of enriching each area and of unifying them all.
Enriched category theory: Categories C enriched over a category V are categories with Hom-sets HomC not just a set or class but with the structure of objects in the category V. Enrichment over V is a way to rise the categorical dimension
Homotopical algebra (published as a book and also sometimes called noncommutative homological algebra): The study of various model categories and the interplay between fibrations, cofibrations and weak equivalences in arbitrary closed model categories
V-categories: A category C with an action ⊗ :V × C → C which is associative and unital up to coherent isomorphism, for V a symmetric monoidal category. V-categories can be seen as the categorification of R-modules over a commutative ring R
Crystalline cohomology: A p-adic cohomology theory in characteristicp invented to fill the gap left by étale cohomology which is deficient in using mod p coefficients for this case. It is sometimes referred to by Grothendieck as the yoga of de Rham coefficients and Hodge coefficients since crystalline cohomology of a variety X in characteristic p is like de Rham cohomology mod p of X and there is an isomorphism between de Rham cohomology groups and Hodge cohomology groups of harmonic forms
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
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.
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.
Clubs (category theory) and coherence (category theory). A club is a special kind of 2-dimensional theory or a monoid in Cat/(category of finite sets and permutations P), each club giving a 2-monad on Cat
1972
John Isbell
Locales: A "generalized topological space" or "pointless spaces" defined by a lattice (a complete Heyting algebra also called a Brouwer lattice) just as for a topological space the open subsets form a lattice. If the lattice possess enough points it is a topological space. Locales are the main objects of pointless topology, the dual objects being frames. Both locales and frames form categories that are each other's opposite. Sheaves can be defined over locales. The other "spaces" one can define sheaves over are sites. Although locales were known earlier John Isbell first named them
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.
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
formulas are terms of type Ω (arrows from types to Ω)
connectives are induced from the internal Heyting algebra structure of Ω
quantifiers bounded by types and applied to formulas are also treated
for each type X there are also two binary relations =X (defined applying the diagonal map to the product term of the arguments) and ∈X (defined applying the evaluation map to the product of the term and the power term of the arguments).
A formula is true if the arrow which interprets it factor through the arrow true:1→Ω. The Mitchell-Bénabou internal language is a powerful way to describe various objects in a topos as if they were sets and hence is a way of making the topos into a generalized set theory, to write and prove statements in a topos using first order intuitionistic predicate logic, to consider toposes as type theories and to express properties of a topos. Any language L also generates a linguistic toposE(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 categoryMR whenever M is a model category. Another advantage of the Reedy structure is that its cofibrations, fibrations and factorizations are explicit. In a Reedy category there is a notion of an injective and a surjective morphism such that any morphism can be factored uniquely as a surjection followed by an injection. Examples are the ordinal α considered as a poset and hence a category. The opposite R° of a Reedy category R is also a Reedy category. The simplex categoryΔ and more generally for any simplicial setX 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
Shows the existence of a global closed model structure on the category of simplicial sheaves on a topological space, with weak assumptions on the topological space
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.
φk : Xk → X1 × X0 ... × X0X1 (induced by X(αi): Xk → X1) assigned to X
is a weak equivalence of simplicial sets for k ≥ 2.
Segal categories are a weak form of S-categories, in which composition is only defined up to a coherent system of equivalences.
Segal categories were defined one year later implicitly by Graeme Segal. They were named Segal categories first by William Dwyer–Daniel Kan–Jeffrey Smith 1989. In their famous book Homotopy invariant algebraic structures on topological spaces, J. Michael Boardman and Rainer Vogt called them quasi-categories. A quasi-category is a simplicial set satisfying the weak Kan condition, so quasi-categories are also called weak Kan complexes
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 categoryhE
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 categoryC 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.
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))
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
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 an ABC (co)fibration category is canonically associated a (left) right Heller derivator. Topological spaces with homotopy equivalences as weak equivalences, Hurewicz cofibrations as cofibrations and Hurewicz fibrations as fibrations form an ABC model category, the Hurewicz model structure on Top. Complexes of objects in an abelian category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations form an ABC precofibration category
Pursuing stacks: Manuscript circulated from Bangor, written in English in response to a correspondence in English with Ronald Brown and Tim Porter, starting with a letter addressed to Daniel Quillen, developing mathematical visions in a 629 pages manuscript, a kind of diary, and to be published by the Société Mathématique de France, edited by G. Maltsiniotis.
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
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
Gives a global closed model structure on the category of simplicial presheaves. John Jardine has also given a model structure in the category of simplicial presheaves
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.
Allegories: An abstraction of the category of sets with relations as morphisms, it bears the same resemblance to binary relations as categories do to functions and sets. It is a category in which one has in addition to composition a unary operation reciprocation R° and a partial binary operation intersection R ∩ S, like in the category of sets with relations as morphisms (instead of functions) for which a number of axioms are required. It generalizes the relation algebra to relations between different sorts.
A∞-categories and A∞-functors: Most commonly in homological algebra, a category with several compositions such that the first composition is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. (associative up to higher homotopy). A stands for associative.
Def: A category C such that
for all X, Y in Ob(C) the Hom-sets HomC(X,Y) are finite-dimensional chain complexes of Z-graded modules
for all objects X1, ..., Xn in Ob(C) there is a family of linear composition maps (the higher compositions)
of degree n − 2 (homological grading convention is used) for n ≥ 1
m1 is the differential on the chain complex HomC(X,Y)
mn satisfy the quadratic A∞-associativity equation for all n ≥ 0.
m1 and m2 will be chain maps but the compositions mi of higher order are not chain maps; nevertheless they are Massey products. In particular it is a linear category.
When there are no higher maps (trivial homotopies) C is a dg-category. Every A∞-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology.
The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of A∞-categories and A∞-functors. Many features of A∞-categories and A∞-functors come from the fact that they form a symmetric closed multicategory, which is revealed in the language of comonads. From a higher-dimensional perspective A∞-categories are weak ω-categories with all morphisms invertible. A∞-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.
Formulates the homological mirror symmetry conjecture: X a compact symplectic manifold with first Chern classc1(X) = 0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(FukX) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of Db(CohY) (the bounded derived category of coherent sheaves on Y).
AST, Algebraic set theory: Also sometimes called categorical set theory. It was developed from 1988 by André Joyal and Ieke Moerdijk, and was first presented in detail as a book in 1995 by them. AST is a framework based on category theory to study and organize set theories and to construct models of set theories. The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded, ...), the various constructions of the cumulative hierarchy of sets, forcing models, sheaf models and realisability models. Instead of focusing on categories of sets AST focuses on categories of classes. The basic tool of AST is the notion of a category with class structure (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a universal object (a universe)) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of ZF-algebras (Zermelo-Fraenkel algebras) and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first-order logic of elementary set theory on the other. The subcategory of sets in a class category is an elementary topos and every elementary topos occurs as sets in a class category. The class category itself always embeds into the ideal completion of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory (BIST) that is logically complete with respect to class category models. Therefore, class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The ZF-axioms are nothing but a description of the free ZF-algebra just as the Peano axioms are a description of the free monoid on one generator. In this perspective the models of set theory are algebras for a suitably presented algebraic theory and many familiar set theoretic conditions (such as well-foundedness) are related to familiar algebraic conditions (such as freeness). Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes.
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.
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.
Cobordism hypothesis (Extended TQFT hypothesis I): The n-category of which n-dimensional extended TQFTs are representations, nCob, is the free stable weak n-category with duals on one object.
Stabilization hypothesis: After suspending a weak n-category n + 2 times, further suspensions have no essential effect. The suspension functor S: nCatk→nCatk+1 is an equivalence of categories for k = n + 2.
Extended TQFT hypothesis II: An n-dimensional unitary extended TQFT is a weak n-functor, preserving all levels of duality, from the free stable weak n-category with duals on one object to nHilb.
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.
Kochen–Specker theorem in topos theory of presheaves: The spectral presheaf (the presheaf that assigns to each operator its spectrum) has no global elements (global sections) but may have partial elements or local elements. A global element is the analogue for presheaves of the ordinary idea of an element of a set. This is equivalent in quantum theory to the spectrum of the C*-algebra of observables in a topos having no points.
Noncommutative schemes: The pair (Spec(A),OA) where A is an abelian category and to it is associated a topological space Spec(A) together with a sheaf of rings OA on it. In the case when A = QCoh(X) for X a scheme the pair (Spec(A),OA) is naturally isomorphic to the scheme (XZar,OX) using the equivalence of categories QCoh(Spec(R)) = ModR. More generally abelian categories or triangulated categories or dg-categories or A∞-categories should be regarded as categories of quasicoherent sheaves (or complexes of sheaves) on noncommutative schemes. This is a starting point in noncommutative algebraic geometry. It means that one can think of the category A itself as a space. Since A is abelian it allows to naturally do homological algebra on noncommutative schemes and hence sheaf cohomology.
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.
Temperley–Lieb categories: Objects are enumerated by nonnegative integers. The set of homomorphisms from object n to object m is a free R-module with a basis over a ring R. R is given by the isotopy classes of systems of (|n| + |m|)/2 simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs |n| points on the bottom and |m| points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley–Lieb categories are categorized Temperley–Lieb algebras.
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.
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.
ASD (Abstract Stone duality): A reaxiomatisation of the space and maps in general topology in terms of λ-calculus of computable continuous functions and predicates that is both constructive and computable. The topology on a space is treated not as a lattice, but as an exponential object of the same category as the original space, with an associated λ-calculus. Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role (an overt object is the dual to a compact object), forming an arithmetic universe (pretopos with lists) with general recursion.
Constructs a model category with certain generalized Segal categories as the fibrant objects, thus obtaining a model for a homotopy theory of homotopy theories. Complete Segal spaces are introduced at the same time.
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).
Influential book: sketches of an elephant – a topos theory compendium. It serves as an encyclopedia of topos theory (two out of three volumes published as of 2008).
Paper A categorical semantics of quantum protocols published that starts the Oxford school of Categorical Quantum Mechanics, based on the theory of compact closed categories.
2004
William Dwyer-Philips Hirschhorn-Daniel Kan-Jeffrey Smith
Quantum categories and quantum groupoids: A quantum category over a braided monoidal categoryV is an object R with an opmorphismh: 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.
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).
Monumental book: Higher topos theory: In its 940 pages Jacob Lurie generalizes the common concepts of category theory to higher categories and defines n-toposes, ∞-toposes, sheaves of n-types, ∞-sites, ∞-Yoneda lemma and proves Lurie characterization theorem for higher-dimensional toposes. Lurie's theory of higher toposes can be interpreted as giving a good theory of sheaves taking values in ∞-categories. Roughly an ∞-topos is an ∞-category which looks like the ∞-category of all homotopy types. In a topos mathematics can be done. In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the (n+1)-category 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. An introduction into this circle of ideas can be found in the Kerodon project.
Sketch of proof of Baez-Dolan tangle hypothesis and Baez-Dolan cobordism hypothesis which classify extended TQFT in all dimensions. Jacob Lurie later publishes the complete proof of the cobordism hypothesis (2010).
First textbook for the emerging field identifying itself as applied category theory, in which category theory is applied outside pure mathematics: An Invitation to Applied Category Theory: Seven Sketches in Compositionality
