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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.
@article{TAC_2021_37_a33,
author = {Peter F. Faul and Graham Manuell and Jos\'e Siqueira},
title = {2-dimensional bifunctor theorems and distributive laws},
journal = {Theory and applications of categories},
pages = {1149--1175},
publisher = {mathdoc},
volume = {37},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2021_37_a33/}
}
TY - JOUR AU - Peter F. Faul AU - Graham Manuell AU - José Siqueira TI - 2-dimensional bifunctor theorems and distributive laws JO - Theory and applications of categories PY - 2021 SP - 1149 EP - 1175 VL - 37 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2021_37_a33/ LA - en ID - TAC_2021_37_a33 ER -
Peter F. Faul; Graham Manuell; José Siqueira. 2-dimensional bifunctor theorems and distributive laws. Theory and applications of categories, Tome 37 (2021), pp. 1149-1175. http://geodesic.mathdoc.fr/item/TAC_2021_37_a33/