## A Universal Characterisation of the Closed Euclidean Interval

Venue: | in: Proceedings of 16th Annual IEEE Symposium on Logic in Computer Science |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Escardo_auniversal,

author = {Martín H. Escardo and Alex K. Simpson},

title = {A Universal Characterisation of the Closed Euclidean Interval},

booktitle = {in: Proceedings of 16th Annual IEEE Symposium on Logic in Computer Science},

year = {},

pages = {115--125}

}

### OpenURL

### Abstract

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. The universal property of an interval object provides a mechanism for defining functions on the interval. We use this to define basic arithmetic operations, and to verify equations between them. It also allows us to develop an analogue of the primitive recursive functions, yielding a natural class of computable functions on the interval. Contents 1