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

@article{Clairambault2014TheBO, 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}, year={2014}, volume={24} }

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…

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## References

SHOWING 1-10 OF 29 REFERENCES

### Locally cartesian closed categories and type theory

- Mathematics, PhilosophyMathematical 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

- Mathematics, Philosophy
- 2004

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

- MathematicsCSL
- 1994

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.

### Fibered categories and the foundations of naive category theory

- PhilosophyJournal of Symbolic Logic
- 1985

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

- Mathematics, PhilosophyTYPES
- 1995

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

- PhilosophyCambridge 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

- MathematicsPOPL '90
- 1989

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

- Computer Science
- 1997

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

- PhilosophyStudies in proof theory
- 1984

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.