The biequivalence of locally cartesian closed categories and Martin-Löf type theories
نویسندگان
چکیده
منابع مشابه
The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories
Seely’s paper Locally cartesian closed categories and type theory contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π,Σ, and extensional identity types. However, Seely’s proof relies on the problematic assumption that substitution in types can be interpreted by pullback...
متن کاملLocally cartesian closed categories and type theory
0. Introduction. I t is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a 'generalized set', for example an 'Aindexed set', is represented by a morphism B^-A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed categor...
متن کاملThe strength of some Martin-Löf type theories
One objective of this paper is the determination of the proof–theoretic strength of Martin– Löf’s type theory with a universe and the type of well–founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with ∆2 comprehension and bar induction. As Martin-Löf intended to formulate a system of co...
متن کاملThe Interpretation of Intuitionistic Type Theory in Locally Cartesian Closed Categories - an Intuitionistic Perspective
We give an intuitionistic view of Seely’s interpretation of Martin-Löf’s intuitionistic type theory in locally cartesian closed categories. The idea is to use Martin-Löf type theory itself as metalanguage, and E-categories, the appropriate notion of categories when working in this metalanguage. As an E-categorical substitute for the formal system of Martin-Löf type theory we use E-categories wi...
متن کاملOn the Interpretation of Type Theory in Locally Cartesian Closed Categories
We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to deene a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematical Structures in Computer Science
سال: 2014
ISSN: 0960-1295,1469-8072
DOI: 10.1017/s0960129513000881