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Naïm Favier (Jul 21 2024 at 15:51):In a general monoidal category with diagonals δ : A → A ⊗ A, is there a reason to expect that δ play well with the associator α : A ⊗ (A ⊗ A) → (A ⊗ A) ⊗ A, in the sense that α ∘ (id ⊗ δ) ∘ δ = (δ ⊗ id) ∘ δ ? Or is that only true in a (semi?)cartesian monoidal category?
Naïm Favier (Jul 21 2024 at 15:53):(say we impose that δ : 1 → 1 ⊗ 1 is an inverse of the unitors)
Matteo Capucci (he/him) (Jul 22 2024 at 07:29):it depends what 'with diagonals' means. if it's just a supply of those maps, I don't think so. if those are the comultiplications of cosemigropus, then yeah, by definition. but the usual coherence proof doesn't go through if those diagonals aren't natural.