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2-dimensional bifunctor theorems and distributive laws

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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.

http://www.tac.mta.ca/tac/volumes/37/34/37-34.pdf

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