2-dimensional bifunctor theorems and distributive laws

Peter F. Faul, Graham Manuell, José Siqueira

In this paper we consider the conditions that need to be satisfied by two families of pseudofunctors with a common codomain for them to be collated into a bifunctor. We observe similarities between these conditions and distributive laws of monads before providing a unified framework from which both of these results may be inferred. We do this by proving a version of the bifunctor theorem for lax functors. We then show that these generalised distributive laws may be arranged into a 2-category Dist(B, C, D), which is equivalent to Lax_op(B,Lax_op(C,D)). The collation of a distributive law into its associated bifunctor extends to a 2-functor into Lax_op(B x C, D), which corresponds to uncurrying via the aforementioned equivalence. We also describe subcategories on which collation itself restricts to an equivalence. Finally, we exhibit a number of natural categorical constructions as special cases of our result.

Keywords: morphism of bicategories, triple, braiding, curry, exponential

2020 MSC: 18D05, 18C15

Theory and Applications of Categories, Vol. 37, 2021, No. 34, pp 1149-1175.

Published 2021-11-25.


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