A cartesian closed category in Martin-Löf's intuitionistic type theory
نویسنده
چکیده
First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to construct the cartesian closed category InfBas, which is equivalent to the category ScDom of Scott Domains. Then, we will show that all the definitions and the proof of all the properties that one needs in order to show that InfBas is indeed a cartesian closed category can be formalized within Martin-Löf’s Intuitionistic Type Theory. Mathematics Subject Classification: 03B15, 03B20.
منابع مشابه
Kripke Semantics for Martin-Löf's Extensional Type Theory
It is well-known that simple type theory is complete with respect to nonstandard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of inte...
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 کاملThe 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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Theor. Comput. Sci.
دوره 290 شماره
صفحات -
تاریخ انتشار 2003