Bifinite Chu Spaces

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

Towards Full Completeness of the Linear Logic of Chu Spaces

We investigate the linear logic of Chu spaces as defined by its dinaturality semantics. For those formulas of multiplicative linear logic limited to at most two occurrences of each variable we prove full completeness of Girard's MIX-free axiomatization, namely that the cut-free proofnets of such formulas are in a natural bijection with the dinatural elements of the corresponding functors. 1 Introduction Whereas ordinary logic axiomatizes theorems, linear logic axiomatizes proofs. The semantic criterion for theoremhood is validity: the truth function denoted by a formula is required to be universally true. Following Lambek and Scott [11] and (as applied to linear logic) Blute and Scott [3, 4], we shall take the semantic criterion for proofhood to be naturality: the transformation denoted by a proof is required to commute with all morphisms of the ambient category. Dinaturality is a small but important generalization of naturality accommodating mixed variance, the possibility of a varia..