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Asad Saeeduddin (Feb 21 2022 at 21:55):Consider the following monoidal structures on the category of sets:
a + b = coproduct
0 = empty set
a * b = product
1 = singleton set
a ⊠ b = (a * b) + (a + b)
0 = empty set
For each monoidal structure M, and for every monoidal category C, there is a category of M-monoidal functors and monoidal natural transformations [C, Set]. Let's call *-monoidal functors "applicatives", +-monoidal functors "alternatives", and ⊠-monoidal functors "alignables".
It seems to me that each applicative and alternative induces an alignable in a relatively obvious way, by utilizing the embedding of the coproduct. I believe these extend to functors Applicative -> Alignable and Alternative -> Alignable.
My question is whether these functors are in fact right adjoints, and therefore admit a notion of the "free applicative/alternative generated by an alignable".
PS: Sorry to not have provided sufficient detail here (I'm planning to come back and flesh this out) but every time I start asking this question I get bogged down in the details and never end up asking it. Figured someone might already have worked all this out and be able to just point me to the details.