Bifinite Chu Spaces

Abstract

This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are

• a characterization of monics in each of the three categories;

• existence (or the lack thereof) of colimits and a characterization of finite objects in each of the corresponding categories using monomorphisms/injections (denoted as iC, iE, and iB, respectively);

• a formulation of bifinite Chu spaces with respect to iC;

• the existence of universal, homogeneous Chu spaces in this category.

Unanticipated results driving this development include the fact that:

• in C, a morphism (f,g) is monic iff f is injective and g is surjective while for E and B, (f,g) is monic iff f is injective (but g is not necessarily surjective);

• while colimits always exist in iE, it is not the case for iC and iB;

• not all finite Chu spaces (considered set-theoretically) are finite objects in their categories.

Chu spaces are a general framework for studying the dualities of objects and properties; points and open sets; terms and types, under rich mathematical contexts, with important connections to several sub-disciplines in computer science and mathematics. Traditionally, the study on Chu spaces had a “non-constructive” flavor. There was no framework in which to study constructions on Chu spaces with respect to their behavior in permitting a transition from finite to infinite in a continuous way and to study which constructs are continuous functors in a corresponding algebroidal category. The work presented here provides a basis for a constructive analysis of Chu spaces and opens the door to a more systematic investigation of such an analysis in a variety of settings.