The superficial similarity between the Chu construction and the Hyland-Tan double glueing construction G has been observed widely. This paper establishes a more formal mathematical relationship between the two.
We show that double glueing on relations subsumes the Chu construction on sets: we present a full monoidal embedding of the category chu(Set,K) of biextensional Chu spaces over K into G(Rel^K), and a full monoidal embedding of the category Chu(Set,K) of Chu spaces over K into IG(Rel^K), where we define IG, the intensional double glueing construction, by substituting multisets for sets in G.
We define a biextensional collapse from IG to G which extends the familiar notion on Chu spaces. This yields a new interpretation of the monic specialisation implicit in G as a form of biextensionality.