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Stream: theory: category theory

Topic: Monoidal vs Commutative Monads

view this post on Zulip Naïm Favier (Feb 18 2024 at 16:12):

I seem to have fallen down this rabbit hole... let me attempt to summarise the situation:

  1. A strong monad T on a monoidal category is a left-strong and right-strong monad such that the two maps (A ⊗ TB) ⊗ C → T (A ⊗ (B ⊗ C)) agree. A commutative monad is a strong monad such that the usual diagram commutes, which does not require a symmetric monoidal category.
  2. If the category is also symmetric monoidal (with c : A ⊗ B → B ⊗ A), then let us say that the monad is symmetrically strong if one strength determines the other, i.e. σ = Tc ∘ τ ∘ c. The nlab page on monoidal monads assumes (remark 2.1) that all strong monads on a symmetric monoidal category are symmetrically strong.
  3. In a monoidal category, there is a one-to-one correspondence between the structures of a commutative monad (with unrelated strengths) and a monoidal monad on a given monad.
  4. Under this correspondence, if the category is symmetric monoidal, then the property of being a symmetrically strong commutative monad corresponds to being a symmetric monoidal monad. Thus, under the nlab's assumption, we recover the correspondence between commutative monads and symmetric monoidal monads.

3 and 4 are basically untangling the symmetry assumptions from the results of Anders Kock (Monads on Symmetric Monoidal Closed Categories and Strong Functors and Monoidal Monads), and would require careful checking but I've pretty much convinced myself they hold by skimming over the proofs and looking at where the assumptions are used.

In particular this means that not all monoidal monads are symmetric: that assumption would be equivalent to the assumption that all commutative monads are symmetrically strong. (That said I have no idea if the counterexample in the linked paper is correct.)