Universality
The category Top of topological spaces and their continuous functions embeds in Chu(Set, 2) in the sense that there exists a full and faithful functor F : Top → Chu(Set, 2) providing for each topological space (X, T) its representation F((X, T)) = (X, ∈, T) as noted above. This representation is moreover a realization in the sense of Pultr and Trnková, namely that the representing Chu space has the same set of points as the represented topological space and transforms in the same way via the same functions.
Chu spaces are remarkable for the wide variety of familiar structures they realize. Lafont and Streicher point out that Chu spaces over 2 realize both topological spaces and coherent spaces (introduced by J.-Y. Girard to model linear logic ), while Chu spaces over K realize any category of vector spaces over a field whose cardinality is at most that of K. This was extended by Vaughan Pratt to the realization of k-ary relational structures by Chu spaces over 2k. For example the category Grp of groups and their homomorphisms is realized by Chu(Set, 8) since the group multiplication can be organized as a ternary relation. Chu(Set, 2) realizes a wide range of ``logical`` structures such as semilattices, distributive lattices, complete and completely distributive lattices, Boolean algebras, complete atomic Boolean algebras, etc. Further information on this and other aspects of Chu spaces, including their application to the modeling of concurrent behavior, may be found at Chu Spaces.
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