Table of Contents
Is homotopy type theory constructive?
Under the propositions as types interpretation, first order logic maps soundly and completely into type theory (Martin-Löf 74, section 3.1, Barendregt 91, section 4). But now the homotopy 0-types in homotopy type theory already give a constructive set theory, natively, without further axiomatization.
What is cubical type theory?
Cubical type theory is a version of homotopy type theory in which univalence is not just an axiom but a theorem, hence, since this is constructive, has “computational content”. Cubical type theory models the infinity-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets, whence the name.
Does homotopy type theory provide a foundation for mathematics?
Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory.
What are higher inductive types?
Higher inductive types (HITs) are a generalization of inductive types which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated identity types.
What is simple type theory?
Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory.
What are inductive data types?
Inductive data type may refer to: Algebraic data type, a datatype each of whose values is data from other datatypes wrapped in one of the constructors of the datatype. Inductive family, a family of inductive data types indexed by another type or value.
Is recursion an induction?
Recursion is the process in which a function is called again and again until some base condition is met. Induction is the way of proving a mathematical statement. 2. It is the way of defining in a repetitive manner.
How many types of theory are there?
Sociologists (Zetterberg, 1965) refer to at least four types of theory: theory as classical literature in sociology, theory as sociological criticism, taxonomic theory, and scientific theory. These types of theory have at least rough parallels in social education. Some of them might be useful for guiding research.
What is n-type homotopy?
Homotopy n -types. A homotopy type that is an n-truncated object in an (∞,1)-category or equivalently that interprets a type of homotopy level n+2 is also called a homotopy n-type or n-type for short. For topological spaces / ∞-groupoids this means that all homotopy groups above degree n are trivial.
What is homotopy type theory?
Homotopy type theory is a flavor of type theory – specifically of intensional ( Martin-Löf -) dependent type theory – which takes seriously the natural interpretation of identity types as formalizing path space objects in homotopy theory.
What is a homotopy between two morphisms?
C in which one does homotopy theory, there is a notion of homotopy between morphisms, which is closely related to the 2-morphisms in higher category theory: a homotopy between two morphisms is a way in which they are equivalent. (\\infty,1) -category.
Is left homotopy a cofibration category?
Clearly the concept of left homotopy in def. 0.5 only needs part of the model category axioms and thus makes sense more generally in suitable cofibration categories. Dually, the concept of path objects in def. 0.5 makes sense more generally in suitable fibration categories such as categories of fibrant objects in the sense of Brown.