A Note on Bainbridge’s Power Set Construction

The category Rel of sets and relations has two natural traced monoidal structures: in Rel Tr , the tensor is given by disjoint union, and in Rel Tr by products of sets. Already in 1976, predating the definition of traced monoidal categories by 20 years, Bainbridge has shown how to model flowcharts and networks in these two respective settings. Bainbridge has also pointed out that one can move from one setting to the other via the power set operation. However, Bainbridge’s power operation is not functorial, and in this paper we show that there is no traced monoidal embedding of Rel Tr into Rel Tr whose object part is given by the power set operation. On the other hand, we show that there is such an embedding whose object part is given by the power-multiset operation. Introduction Predating the definition of traced monoidal categories [2] by 20 years, Bainbridge [1] has pointed out in 1976 that there exist (in today’s terminology) two natural traced monoidal structures on the category Rel of sets and relations. The first one is Rel Tr , where the tensor product is given by disjoint union of sets. The second one is Rel Tr , where tensor is given by products of sets. Bainbridge used these categories to give a compositional semantics to flowcharts and networks, respectively, and he pointed out a duality between the two situations: the power set operation takes the first category to the second, and it gives rise to a homset-wise Galois connection. Bainbridge’s power operation maps a set to the power set , and a relation "! to the relation # $ " #! given by %& # (' iff for all )+*&% , ), .implies -/*&' . Remarkably, this operation preserves not only composition and tensor, but also trace. However, it does not preserve identities, and it is therefore not a functor. One may now ask whether there is some variant of Bainbridge’s construction that yields an actual functor of traced monoidal categories. More precisely: is there a traced monoidal embedding of Rel Tr into Rel 0 Tr 1 whose object part is given by power sets? The answer, as we shall see, is no. In fact, there is no traced monoidal embedding between these categories that maps finite sets to finite sets. On the other hand, we will show that such an embedding exists whose object part is given by the power-multiset operation. Thanks to Thomas Hildebrandt for pointing out a mistake in the first draft of this manuscript. An embedding of 2 Rel 35463 Tr 7 into 2 Rel 398:3 Tr ;<7 Let Rel be the category of sets and relations, and let Relfin be the full subcategory of finite sets. On Rel, we consider two traced monoidal structures Rel = Tr and Rel 0 Tr . For the first one, is disjoint union of sets, and for " 9 > @?> A!B @? , Tr CD " E F A! is given by )G Tr CH # <iff there exist IKJ MLNLNLO PI QR*S? , with TVUXW , such that )Y IZJ[ \LMLNLP I Q] .. The second traced monoidal structure is given by as the product of sets, and for