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The Legendre-Fenchel transform from a category theoretic perspective

The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this paper we show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty]$. A key ingredient is Pavlovic's 'nucleus of a profunctor' construction. The pairing between a vector space and its dual can be viewed as an $\overline {\mathbb{R}}$-profunctor; the construction of the nucleus of this profunctor is the construction of a lot of the theory of the Legendre-Fenchel transform. For a relation between sets viewed as a $\{\mathrm{true},\mathrm{false}\}$-valued profunctor, the construction of the nucleus is the construction of the Galois connection associated to the relation. One insight given by this approach is that the relevant structure on the function spaces involved in the Legendre-Fenchel transform is something like a metric but is asymmetric and can take negative values. This '$\overline {\mathbb{R}}$-structure' is a considerable refinement of the usual partial order on real-valued function space and it allows a natural interpretation of Toland-Singer duality and of the two tropical module structures on the set of convex functions.