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Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.
added pointer to
Beno Eckmann, Peter Hilton, Structure maps in group theory, Fundamenta Mathematicae 50 (1961), 207-221 (doi:10.4064/fm-50-2-207-221)
Beno Eckmann, Peter Hilton, Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)
added pointer now also to Eckmann-Hilton’s part III
Have not added pointer yet to part II. That part is interesting in that it lays out a lot of basic theory of limits and colimits, which apparently was new at that time (and hence maybe should be cited at limit or so) but part II does not seem to talk much about group objects or internalization as such. (Or maybe I missed it, have only skimmed through it, to be frank.)
added pointer to:
added these two references, apparently preceding Eckmann-Hilton:
Alexander Grothendieck, p. 340 (3 of 23) in: Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf)
Alexander Grothendieck, p. 104 (7 of 21) in: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)
added pointer to:
No wait. Where are you taking this idea from?
In this case your habit of saying “one could define” happens to be appropriate (maybe unintentionally?): One could, but one does not.
What you propose is, in paraphrase, to define a group object in a non-cartesian monoidal category to be a monoid object $G$ such that its global elements $I \to G$ (only) have inverses. This would yield a rather loose notion of group objects. Are there actually any use-cases of this would-be notion?
What one actually does when discussing group objects in general monoidal categories is to postulate a compatible comonoid object-structure on the given monoid (whose coproduct substitutes for the missing diagonal maps). Then the inverse-assigning map is called an antipode and the whole thing is traditionally called a Hopf monoid.
I have added a Remark (here) to this extent. It still mentions your “could be”-definition in the last sentence, but unless we have any examples of use-cases, I’d be inclined to delete this.
This $G(S)$ is shorthand for $y(G)(S) = Hom_C(S,G)$, i.e. for regarding $G$ as a presheaf (the one represented by $G$) and as such evaluated on $S$.
I can’t edit right now. If nobody else does I’ll add clarification to the entry tomorrow.
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