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I have edited the second point under examples on the cogroup page. I replaced what I believe to be an erroneous $\operatorname{hTop}$ with $\operatorname{hTop}_*$, and have included a reference for the claim that there are cogroups in $\operatorname{hTop}_*$ which are not suspensions.
Thanks, Michael. You might already know this, but it is convenient to link to nLab pages like cogroup by placing the page name between double brackets, like this: [[cogroup]].
I added some minor edits at cogroup, including a warning about possible terminological clashes when applying “co” to the concept of monoid.
Is it really a terminological “clash” if one usage is strictly more general than the other?
If you can suggest a better word than “clash”, please feel free. I think it becomes a clash after you specialize to a monoidal product different from coproduct, and in any case I don’t think a remark is out of order. “Potential clash”, perhaps?
Edit: The opening sentence had “There can be terminological clashes” which I went ahead and changed to “There are potential terminological clashes”. Which I think is saying the exact same thing, but please go ahead and change again if you want.
I would just say “note that though cogroup objects make sense only in cocartesian monoidal categories, comonoids make sense in arbitrary monoidal categories. This is because the axioms of a group involve duplication of variables, while the axioms of a monoid don’t.”
(Ok, true, the notion of cogroup object defined on that page doesn’t require coproducts to exist in the category, but it reduces to the case when coproducts exist by using the co-Yoneda embedding. Similarly, a monoid also makes sense in a multicategory, but any multicategory can be embedded in a monoidal category.)
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