Large analytic [0,1]-duality and its combinatorial variants

Duality in mathematics can often be described by means of *-autonomous categories, or symmetric monoidal categories with a dualising object. Such categories can be roughly classified into "small" (consisting of finite-dimensional or finitely-generated objects) and "large", the latter being more unusual and more interesting. We briefly explain this distinction by presenting a series of well-known examples, including the *-autonomous category (k-vect) of finite-dimensional vector spaces, the category (Rel) of relations between sets, the category (CSLat) of complete semilattices and the category (CohGr) of coherence graphs. After that, we present a new general construction of large *-autonomous categories by localisation of suitable closed symmetric monoidal categories C, defined with the aid of certain "localising objects" in these categories. We illustrate this construction by introducing the monoidal category (Bp) of completely additive blueprints, which generalise an idea originally due to Oliver Lorscheid, and then apply the general localisation construction to two-point blueprint {0,1}, obtaining a *-autonomous category equivalent to the category (CohGr) of coherence graphs; as a corollary, we prove that (CohGr) is complete and cocomplete. We state a new duality theorem, which asserts that other completely additive blueprints, such as the unit segment [0,1] or the finite sets of integers {0,1,2,...,r-1}, also are localising objects in category (Bp), and therefore define new large *-autonomous categories by localisation. We illustrate this theorem by measure-theoretical and combinatorial examples that arise from [0,1]- and {0,1,...,r-1}-duality, respectively, explain why these results are surprising, and outline the general strategy of the proof of the main theorem.