![]() Under this proofs as programs-paradigm, type theory is a mathematical formalization of a programming language. Since such a proof is constructive, the term witnessing it being a concrete implementation, and since type theory strictly works by rewriting rules, one may identify the construction of a term in type theory as a program whose output is a certain type. This identification leads to a very fruitful identification of operations on types with logical operations. Under this propositions are types-paradigm a proof of the proposition is nothing but a term of the corresponding type. On the one hand, logic itself is subsumed in the plain idea of operations on terms of types, by observing that any type X X may be thought of as the type of terms satisfying some proposition. What may seem like a triviality on first sight turns out to have deep implications:įoundations of mathematics. ![]() Type theory is a branch of mathematical symbolic logic, which derives its name from the fact that it formalizes not only mathematical terms â such as a variable x x, or a function f f â and operations on them, but also formalizes the idea that each such term is of some definite type, for instance that the type â \mathbb between natural numbers.Įxplicitly, type theory is a formal language, essentially a set of rules for rewriting certain strings of symbols, that describes the introduction of types and their terms, and computations with these, in a sensible way. Extensional and intensional function types.Semantics of type-theoretical foundations.Logic versus type theory in categorical semantics.As a formal language for category theory.natural numbers structure, axiom of infinity.powerset structure, axiom of power sets.membership relation, propositional equality, axiom of extensionality.Univalence, function extensionality, internal logic of an (â,1)-topos Homotopy type theory, homotopy type theory - contents ![]() Modal type theory, monad (in computer science)ĭomain specific embedded programming language N-image of morphism into terminal object/ n-truncationĭiagonal function/ diagonal subset/ diagonal relationīishop set/ setoid with its pseudo-equivalence relation an actual equivalence relation Indexed cartesian product (of family of subsingletons)ĭependent product (of family of subterminal objects)ĭependent product type (of family of h-propositions) Subterminal object/ (-1)-truncated object The other zigzag identity for hom-tensor adjunction One of the zigzag identities for hom-tensor adjunction Propositions as types + programs as proofs + relation type theory/category theory logicĬomposition of classifying morphisms / pullback of display maps
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