# Information Systems with Witnesses: The Function Space Construction

## TL;DR

This paper proves that the category of information systems with witnesses and approximable mappings is Cartesian closed, providing a direct proof and showing the natural information system structure of mappings.

## Contribution

It offers a direct proof of the Cartesian closure of the category of information systems with witnesses, enhancing understanding of their algebraic structure.

## Key findings

- The category of information systems with witnesses is Cartesian closed.
- Approximable mappings form a natural information system structure.
- Provides a direct proof of Cartesian closure, simplifying previous approaches.

## Abstract

Information systems with witnesses have been introduced in [D. Spreen. Generalised information systems capture L-domains. arXiv:1610.02260v2] as a logic-style representation of L-domains: The category of such information systems with approximable mappings as morphisms is equivalent to the category of L-domains with Scott continuous functions, which is known to be Cartesian closed. In the present paper a direct proof of the Cartesian closure of the category of information systems with witnesses and approximable mapppings is given. As is shown, the collection of approximable mappings between two information systems with witnesses comes with a natural information system structure.