## Deriving Category Theory from Type Theory (1993)

### BibTeX

@MISC{Crole93derivingcategory,

author = {Roy L. Crole},

title = {Deriving Category Theory from Type Theory},

year = {1993}

}

### OpenURL

### Abstract

This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories. It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics. We begin by showing how the concept of a category can be derived from some simple and primitive mechanisms of monadic type theory. We then show how the notion of a category with finite products can model the most fundamental syntactical constructions of (algebraic) type theory. The idea of naturality is shown to capture, in a syntax free manner, the notion of substitution, and therefore provides a syntax free coding of a multiplicity of type theoretical constructs. Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed λ-calculus with binary products and a unit type. This article provides a new presentation of some old ideas. It is intended to be a tutorial paper aimed at audiences interested in elementary categorical type theory. Further details can be found in [Cro93]. 1 1

### Citations

847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...ples which underlie functional programming. In fact the foundations of λ-calculus were laid many years ago by logicians, but this will not concern us here. The interested reader might care to consult =-=[Chu40]-=- and [Chu41]. Many computer scientists are also aware that the λ-calculus has a formal connection with the notion of a cartesian closed category; this idea is due to Lambek [Lam80]. However, experienc... |

394 |
Category Theory for Computing Science
- Barr, Wells
- 1999
(Show Context)
Citation Context ...ic Type Theory Readers not familiar with type theory will find that [NPS90] is an excellent reference. There are a number of textbooks now available which cover basic category theory; see for example =-=[BW90]-=- or [Pie91]. Before we embark on our derivation of a category, let us review the traditional set-theoretic semantics which can be given to elementary type theory. The “elementary” type theory which we... |

320 |
The Calculi of Lambda Conversion
- Church
- 1941
(Show Context)
Citation Context ...nderlie functional programming. In fact the foundations of λ-calculus were laid many years ago by logicians, but this will not concern us here. The interested reader might care to consult [Chu40] and =-=[Chu41]-=-. Many computer scientists are also aware that the λ-calculus has a formal connection with the notion of a cartesian closed category; this idea is due to Lambek [Lam80]. However, experience shows that... |

215 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...category induced by a finite product. 1 Here we assume that the reader has some knowledge of categories with finite products. The definition can be found in any basic book on category theory, such as =-=[Mac71]-=- or [BW90]. Using such a category we can give a semantics to algebraic type theory as follows: Let C be a category with finite products and let Sg be an algebraic signature. A structure, M, in C for S... |

172 |
Basic Category Theory for Computer Scientists
- Pierce
- 1991
(Show Context)
Citation Context ...eory Readers not familiar with type theory will find that [NPS90] is an excellent reference. There are a number of textbooks now available which cover basic category theory; see for example [BW90] or =-=[Pie91]-=-. Before we embark on our derivation of a category, let us review the traditional set-theoretic semantics which can be given to elementary type theory. The “elementary” type theory which we shall cons... |

18 |
Categories for Types. Cambridge Mathematical Textbooks
- Crole
- 1993
(Show Context)
Citation Context ... This article provides a new presentation of some old ideas. It is intended to be a tutorial paper aimed at audiences interested in elementary categorical type theory. Further details can be found in =-=[Cro93]-=-. 11 Introduction Typed λ-calculus is a subject very well understood by today’s computer scientists, being an embodiment of the basic principles which underlie functional programming. In fact the fou... |

12 | Programming Metalogics with a Fixpoint Type
- Crole
- 1992
(Show Context)
Citation Context ...finite products. The methodology of manipulating naturality and soundness equations to compute general categorical structures can be applied to less well known type theories. Examples can be found in =-=[Cro91]-=-. I would like to thank Andrew Pitts for discussions which threw light on my understanding of the way category-theoretic ideas capture slickly the essence of intricate syntactic constructions. I would... |

11 |
From λ-calculus to Cartesian closed categories
- Lambek
- 1980
(Show Context)
Citation Context ...ght care to consult [Chu40] and [Chu41]. Many computer scientists are also aware that the λ-calculus has a formal connection with the notion of a cartesian closed category; this idea is due to Lambek =-=[Lam80]-=-. However, experience shows that this connection is often only fully appreciated by those working in very theoretical areas of computer science. We hope that this article will go some way towards brid... |