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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 such that its global elements (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 is shorthand for , i.e. for regarding as a presheaf (the one represented by ) and as such evaluated on .
I can’t edit right now. If nobody else does I’ll add clarification to the entry tomorrow.
The Idea section contains
Given a non-cartesian [[monoidal category]] one can still make sense of group objects in the [[formal duality|dual]] guise of [[Hopf monoids]], see there for more and see Rem. \ref{GroupObjectsInGeneralMonoidalCategories} below.
GroupObjectsInGeneralMonoidalCategories is undefined.
Thanks for the alert. Looking through the page history, the link was broken in revision 54 by John Baez.
Am taking care of the entry now…
Okay, so I have fixed the broken link by rearranging a little.
Moreover, I turned the idea alluded to in the section “In a monoidal category” into a Proposition+Proof under “In a cartesian monoidal category”, now here.
The comment that this might be a useful way for thinking about group objects in non-cartesian monoidal categories can maybe wait until there is an example to back this up.
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