A Quillen Model Structure for Chu Spaces

Barr introduced Chu categories as a general construction for generating ∗-autonomous categories, the basic framework for the semantics of Girard's linear logic. Barr singles out two classes of objects in a Chu category for special consideration, the separated and extensional objects. It is shown in [Michael Barr. The separated extensional Chu category. Theory Appl. Categ., 4:No. 6, 137–147 (electronic), 1998] that, under certain circumstances, one can induce a ∗-autonomous structure on the full subcategory of these objects. The manner in which this is done, and the nature of the hypotheses involved, suggest the existence of a homotopy-theoretic interpretation of these ideas. In this paper, we show that this is indeed the case. In particular, we show that it is possible to put a Quillen model structure on certain Chu categories in such a way that a Chu space is separated if and only if it is fibrant, and extensional if and only if it is cofibrant.