Bifinite Chu Spaces

TL;DR

This paper analyzes colimits of finite Chu spaces and their variants, characterizing finite objects and establishing the existence of universal, homogeneous Chu spaces, with implications for linear logic models.

Contribution

It provides a detailed categorical analysis of bifinite Chu spaces, including their monomorphisms, colimits, and the existence of universal homogeneous spaces.

Findings

Colimits exist only in the extensional case.

Monomorphisms differ across Chu space variants.

Universal, homogeneous Chu spaces are constructed.

Abstract

This paper studies colimits of sequences of finite Chu spaces and their ramifications. Besides generic Chu spaces, we consider extensional and biextensional variants. In the corresponding categories we first characterize the monics and then the existence (or the lack thereof) of the desired colimits. In each case, we provide a characterization of the finite objects in terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with respect to the monics of generic Chu spaces, and universal, homogeneous Chu spaces are shown to exist in this category. Unanticipated results driving this development include the fact that while for generic Chu spaces monics consist of an injective first and a surjective second component, in the extensional and biextensional cases the surjectivity requirement can be dropped. Furthermore, the desired colimits are only guaranteed to exist in the extensional case. Finally, not all finite Chu spaces (considered set-theoretically) are finite objects in their categories. This study opens up opportunities for further investigations into recursively defined Chu spaces, as well as constructive models of linear logic.