Distributive laws of set functors over the powerset monad (also known as Kleisli laws for the powerset monad) are well-known to be in one-to-one correspondence with extensions of set functors to functors on the category of sets and relations. We study the question of existence and uniqueness of such distributive laws. Our main result entails that an accessible set functor admits a distributive law over the powerset monad if and only if it preserves weak pullbacks, in which case the so-called power law (which induces the Barr extension) is the unique one. Furthermore, we show that the powerset functor admits exactly three distributive laws over the powerset monad, revealing that uniqueness may fail for non-accessible functors.

翻译：集合函子对幂集幺半群的分配律（亦称幂集幺半群的Kleisli律）众所周知与集合函子向集合及关系范畴上函子的扩张存在一一对应关系。本文研究此类分配律的存在性与唯一性问题。我们的主要结果表明：一个可达集合函子允许幂集幺半群上的分配律当且仅当该函子保持弱拉回，此时所谓的幂律（其诱导Barr扩张）是唯一的分配律。此外，我们证明幂集函子恰好允许三个幂集幺半群上的分配律，这表明对于非可达函子而言唯一性可能不成立。