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Is there a monoidal structure on the category of groups such that the forgetful functor from (Ab,⊗,Z) is lax monoidal?
My first guess is the coproduct could work. The coproduct in Grp is the free product. Universal property of coproducts could give the laxator.
Yes, functors to cartesian monoidal categories are canonically oplax, and to cocartesian monoidal categories canonically lax.
ah, I see, maybe this is the wrong question then
I wanted a monoidal structure on such that an abelian group becomes a -category with one object (by Eckmann-Hilton). Then the forgetful functor should be at least lax and the induced functor restricted to one-object-categories would send a ring to the underlying abelian group.
But aren't these answers answering a different question, where the monoidal structure on Ab is direct sum, while the original question is about tensor?
There does seem to be a "nonabelian tensor product" of groups. I don't know anything about it, but I bet that it makes the inclusion of into strong monoidal, just by restricting to the usual tensor product on Ab.
Amar Hadzihasanovic said:
Yes, functors to cartesian monoidal categories are canonically oplax, and to cocartesian monoidal categories canonically lax.
Ouff, I misremembered the result; it's functors from semicartesian (unit is terminal) to cartesian that are canonically oplax, and dually from semi-cocartesian (unit is initial) to cocartesian that are lax. with tensor product is neither, so the result doesn't apply.