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If C is a closed symmetric monoidal category, the Chu category Chu(C, g) overC and an object g of it was defined by Chu, as a *-autonomous categorygenerated from C. Bishop introduced the category of complemented subsets of aset, in order to overcome the problems generated by the use of negation inconstructive measure theory. Shulman mentions that Bishop's complementedsubsets correspond roughly to the Chu construction. In this paper we explainthis correspondence by showing that there is a Chu representation (a fullembedding) of the category of complemented subsets of a set X into Chu(Set, X xX). A Chu representation of the category of Bishop spaces into Chu(Set, R) isshown, as the constructive analogue to the standard Chu representation of thecategory of topological spaces into Chu(Set, 2). In order to represent thecategory of predicates (with objects pairs (X, A), where A is a subset of X,and the category of complemented predicates (with objects pairs (X, A), where Ais a complemented subset of X, we generalise the Chu construction by definingthe Chu category over a cartesian closed category C and an endofunctor on C.Finally, we introduce the antiparallel Grothendieck construction over a productcategory and a contra-variant Set-valued functor on it of which the Chuconstruction is a special case, in case C is a locally small, cartesian closedcategory.

Chu representations of categories related to constructive mathematics