Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole.
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems.
Vladimir Alexandrovich Voevodsky was a Russian-American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the univalent foundations of mathematics and homotopy type theory.
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Einstein, J. Robert Oppenheimer, Hermann Weyl, John von Neumann, and Kurt Gödel, many of whom had emigrated from Europe to the United States.
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.
In computer science and mathematical logic, a function type is the type of a variable or parameter to which a function has or can be assigned, or an argument or result type of a higher-order function taking or returning a function.
The vertical bar, |, is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke, pipe, bar, or, vbar, and others.
In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression. A product of types is a direct product of two or more types.
In type theory, a system has inductive types if it has facilities for creating a new type from constants and functions that create terms of that type. The feature serves a role similar to data structures in a programming language and allows a type theory to add concepts like numbers, relations, and trees. As the name suggests, inductive types can be self-referential, but usually only in a way that permits structural recursion.
In type theory, an empty type or absurd type, typically denoted is a type with no terms. Such a type may be defined as the nullary coproduct. It may also be defined as the polymorphic type
Steven M. Awodey is an American mathematician and logician. He is a Professor of Philosophy and Mathematics at Carnegie Mellon University.
Thierry Coquand is a French computer scientist and mathematician who is currently a professor of computer science at the University of Gothenburg, having previously worked at INRIA. He is known for his work in constructive mathematics, especially the calculus of constructions.
In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces.
Conor McBride is a Reader in the department of Computer and Information Sciences at the University of Strathclyde. In 1999, they completed a Doctor of Philosophy (Ph.D.) in Dependently Typed Functional Programs and their Proofs at the University of Edinburgh for their work in type theory. They formerly worked at Durham University and briefly at Royal Holloway, University of London before joining the academic staff at the University of Strathclyde.
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.
Thomas Streicher is an Austrian mathematician who is a Professor of Mathematics at Technische Universität Darmstadt. He received his PhD in 1988 from the University of Passau with advisor Manfred Broy.
Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by "categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory.
Michael "Mike" Shulman is an American associate professor of mathematics at the University of San Diego who works in category theory and higher category theory, homotopy theory, logic as applied to set theory, and computer science.
In type theory, a polynomial functor is a kind of endofunctor of a category of types that is intimately related to the concept of inductive and coinductive types. Specifically, all W-types are initial algebras of such functors.
Mikhail Kapranov, is a Russian mathematician, specializing in algebraic geometry, representation theory, mathematical physics, and category theory. He is currently a professor of the Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo.
