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Stream: learning: questions

Topic: monoidal category Grp

Daniel Teixeira (Oct 12 2022 at 19:14):

Is there a monoidal structure on the category of groups such that the forgetful functor from (Ab,⊗,Z) is lax monoidal?

Joe Moeller (Oct 12 2022 at 20:05):

My first guess is the coproduct could work. The coproduct in Grp is the free product. Universal property of coproducts could give the laxator.

Amar Hadzihasanovic (Oct 12 2022 at 20:29):

Yes, functors to cartesian monoidal categories are canonically oplax, and to cocartesian monoidal categories canonically lax.

Daniel Teixeira (Oct 13 2022 at 14:51):

ah, I see, maybe this is the wrong question then

Daniel Teixeira (Oct 13 2022 at 14:51):

I wanted a monoidal structure on $\mathsf{Grp}$ such that an abelian group becomes a $\mathsf{Grp}$-category with one object (by Eckmann-Hilton). Then the forgetful functor $\mathsf{Ab}\to \mathsf{Grp}$ should be at least lax and the induced functor $\mathsf{AbCat}\to \mathsf{GrpCat}$ restricted to one-object-categories would send a ring to the underlying abelian group.

Tobias Fritz (Oct 13 2022 at 15:16):

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 $(\mathrm{Ab},\otimes)$ into $\mathrm{Grp}$ strong monoidal, just by restricting to the usual tensor product on Ab.

Amar Hadzihasanovic (Oct 13 2022 at 15:28):

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. $\mathbf{Ab}$ with tensor product is neither, so the result doesn't apply.