Noncommutative Products on Categories and Chu Construction

The category of Chu spaces is constructed from a given symmetric monoidal closed category 𝐾 and a fixed object in it. If the object is not fixed, we obtain the category 𝐶ℎ𝑢(𝐾), whose objects are Chu spaces and whose morphisms are defined in a more general way. E.E. Skurikhin defined and studied the category of 𝒯 -Chu spaces associated with an arbitrary functor 𝒯 from the product of categories 𝑁 𝑜𝑝 and 𝑀 to the category of sets. In this paper, we prove that if the functor 𝒯 is closed, then the category 𝑀 is isomorphic to a reflective subcategory of the category of 𝒯 -Chu spaces. In the case where the categories 𝑁 and 𝑀 are complete, we present constructions of limits and colimits of arbitrary functors with values in the category of 𝒯 -Chu spaces. We also prove that if 𝐾 is a symmetric monoidal closed category, then 𝐶ℎ𝑢(𝐾) admits the structure of a closed right-monoidal category. As a consequence of the results on 𝒯 -Chu spaces, it follows that the category 𝐶ℎ𝑢(𝐾), as well as the categories of Chu spaces over it, are complete and cocomplete whenever 𝐾 is.