The biequivalence of locally cartesian closed categories and Martin-Löf type theories

  title={The biequivalence of locally cartesian closed categories and Martin-L{\"o}f type theories},
  author={Pierre Clairambault and Peter Dybjer},
  journal={Mathematical Structures in Computer Science},
Seely's paper Locally cartesian closed categories and type theory (Seely 1984) 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 pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou–Hofmann… 

An interpretation of dependent type theory in a model category of locally cartesian closed categories

This paper shows that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory, and constructs a model category of lcc sketches, from which the latter is the model ofdependent type theory.

Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)

It is shown that a version of Martin-Lof type theory with extensional identity, a unit type N1, Sigma, Pi, and a base type is a free category with families both in a 1- and a 2-categorical sense and equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic.

Cartesian closed bicategories: type theory and coherence

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type

$\infty$-type theories

We introduce ∞ -type theories as an ∞ -categorical generalization of the categorical definition of type theories introduced by the second named author. We establish analogous results to the previous


We extend the theory of Artin gluing to strict dependent type theory: given a ƒat functor C E F from the category of contexts of a model of Martin-Löf type theory into a Grothendieck topos E, we may

A continuous computational interpretation of type theories

This thesis provides a computational interpretation of type theory validating Brouwer’s uniform-continuity principle that all functions from the Cantor space to natural numbers are uniformly

Categories with Families: Unityped, Simply Typed, and Dependently Typed

Several equivalence and biequivalence theorems are proved between cwf-based notions and basic notions of categorical logic, such as cartesian operads, Lawvere theories, categories with finite products and limits, cartesian closed categories, and locally cartesian open categories.

A constructive manifestation of the Kleene-Kreisel continuous functionals

Dependent Cartesian Closed Categories

DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax of Martin-L\"{o}f type theory (MLTT) by capturing the categorical counterpart of the generalization of the simply-typed lambda-calculus to MLTT in syntax.



Locally cartesian closed categories and type theory

  • R. Seely
  • Mathematics, Philosophy
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1984
It 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

Decidability of Equality in Categories with Families

Categories with families (cwfs) is a theory which was introduced to be a categorical model of dependently-typed λ-calculus (LF). We first define both theories and discuss about possible variations of

On the Interpretation of Type Theory in Locally Cartesian Closed Categories

It is shown how to construct a model of dependent type theory from a locally cartesian closed category (lccc) that allows to define a semantic function interpreting the syntax of type theory in an lccc.

An Intuitionistic Theory of Types: Predicative Part

Fibered categories and the foundations of naive category theory

Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives: to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent with a well-established and “safe” theory.

Internal Type Theory

We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also

Practical Foundations of Mathematics

  • P. Taylor
  • Philosophy
    Cambridge studies in advanced mathematics
  • 1999
The aim of the book is to exhibit and study the mathematical principles behind logic and induction as needed and used for the formalisation of (the main parts of) Mathematics and Computer Science.

Explicit substitutions

The λ&sgr;-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λ&sgr;-calculus provides a setting for studying the theory of substitutions, with pleasant

Semantics and Logics of Computation: List of Contributors

This volume is based on a summer school at the Isaac Newton Institute and consists of a sequence of linked lecture course by international authorities in the area of semantics and logics of computation.

Intuitionistic type theory

These lectures were given in Padova and Munich later in the same year as part of the meeting on Konstruktive Mengenlehre und Typentheorie which was organized in Munich by Prof. Helmut Schwichtenberg.