Iterated Covariant Powerset is not a Monad

No-Go Theorems for Distributive Laws

“…The negative result for P • P ⇒ P • P was already shown by [12], and can also be recovered from both Theorems 3.6 and 4.20. The other negative results follow from either Theorem 4.15 or Theorem 4.20.…”

“…In order to do this, we isolate sufficient conditions on algebraic theories inducing two monads, such that there can be no distributive law between them. Earlier generalizations of this counterexample have appeared in [12,14]. All the existing approaches involve direct calculations with the distributive law axioms, leading to somewhat opaque conditions.…”

“…These methods are highly valuable, for as stated in [10]: "It can be rather difficult to prove the defining axioms of a distributive law." In fact, it can be so difficult that on occasion a distributive law has been published which later turned out to be incorrect; see [12] for an overview of such cases involving the powerset monad.…”

“…The proofs in this section require different types of substitutions, and this difference impacts their scope of application. The proofs of Theorems 3.6 and 3.13 only require variable-for-variable substitutions, and actually preclude the existence of distributive laws for pointed endofunctors, generalizing[12, Theorem 2.4]. The proof of Theorem 3.15 requires more complex substitutions, implicitly assuming the multiplication axioms.…”

Lattices do not distribute over powerset

“…Nevertheless, distributive laws of a functor that does not preserve weak pullbacks over the powerset are rare and the answer might depend on the kind of axioms we impose over such distributive law. For instance, it is shown in [6] that there is no distributive law of (P, η) over (P, η), correcting a mistake that appeared a couple of times in the literature claiming the opposite [6,Section 5], but, on the other hand, distributive laws of (P, η, μ) over P and distributive laws of P over (P, η, μ) do exist (just take P η • μ and ηP • μ, respectively).…”