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Stream: learning: questions

Topic: monoidal category Grp


view this post on Zulip 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?

view this post on Zulip 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.

view this post on Zulip Amar Hadzihasanovic (Oct 12 2022 at 20:29):

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

view this post on Zulip Daniel Teixeira (Oct 13 2022 at 14:51):

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

view this post on Zulip Daniel Teixeira (Oct 13 2022 at 14:51):

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

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

view this post on Zulip 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. Ab\mathbf{Ab} with tensor product is neither, so the result doesn't apply.