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Motivation: One of the reason that the category of abelian groups is nice is that, unlike the category of groups, it has a monoidal closed structure.

Are there other full subcategories of $\textbf{Grp}$ that aren't a full subcategory of $\textbf{Ab}$ and admit a monoidal closed structure? Is a classification of those subcategories known?

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    $\begingroup$ There are two degrees of freedom here: the monoidal structure and the purported internal hom. Now, certainly Grp is Cartesian, but can't be Cartesian closed. How many other monoidal structures are there on groups? I suspect: not many if you regard groups as one-object groupoids. $\endgroup$
    – fosco
    Aug 23, 2023 at 12:33
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    $\begingroup$ The "funny" tensor product of groups regarded as one-object groupoids is their coproduct. I think with a similar argument you can prove that the _co_Cartesian monoidal structure on Grp can't be closed. And over Cat there aren't other interesting/closed tensor products... $\endgroup$
    – fosco
    Aug 23, 2023 at 12:35
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    $\begingroup$ Not an answer to your question as stated, but instead of going smaller you can go bigger: the $2$-category of groupoids is not only closed monoidal but even cartesian closed. The internal hom here is the functor category. $\endgroup$ Aug 23, 2023 at 17:10
  • $\begingroup$ perks of having the empty groupoid! That's what makes all the difference between Grp and Gpd $\endgroup$
    – fosco
    Aug 23, 2023 at 21:08

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