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nLab > Latest Changes: group object

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2009

    Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 21st 2021
    • (edited Mar 21st 2021)

    added pointer to

    diff, v26, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021

    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.)

    diff, v27, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2021

    added pointer to:

    diff, v31, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2021

    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)

    diff, v33, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    fixed the formatting of some of the old diagrams here,

    added mentioning of the example of simplicial groups

    diff, v35, current

  7. rather than overloading 1

    Nathan Brader

    diff, v40, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2023

    added (here) the type-theoretic expression for the type of group data strcutures

    diff, v41, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023

    added pointer to:

    diff, v44, current

  10. Added the generalization of the concept of group object from cartesian monoidal categories to any monoidal category

    Anonymouse

    diff, v46, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2023
    • (edited Oct 6th 2023)

    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 GG such that its global elements I→GI \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.

    diff, v47, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2023

    Anonymouse, since I gather from your reaction in another thread (here) that you have seen my message above, I assume you silently agree. So I went ahead and deleted the last sentence concerning your purported definition.

    diff, v49, current

    • CommentRowNumber13.
    • CommentAuthorzuoke
    • CommentTimeFeb 15th 2024
    Hi everyone. I can't figure out what dose G(S) in the section "In terms of presheaves of groups" mean. As S and G are both objects in C, why can G be applied to S? Thanks in advance
    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2024
    • (edited Feb 15th 2024)

    This G(S)G(S) is shorthand for y(G)(S)=Hom C(S,G)y(G)(S) = Hom_C(S,G), i.e. for regarding GG as a presheaf (the one represented by GG) and as such evaluated on SS.

    I can’t edit right now. If nobody else does I’ll add clarification to the entry tomorrow.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2024

    Okay, I have expanded the text a little, for clarification (around here).

    diff, v51, current

    • CommentRowNumber16.
    • CommentAuthorJohn Baez
    • CommentTimeAug 14th 2024

    Added concept of group object in a monoidal category; gave Hopf mnonoids their own section.

    diff, v54, current

  17. \mathrm{e} is a morphism from the terminal object to GG. The correct morphism from GG to GG here is id.

    AD

    diff, v57, current

  18. \mathrm{e} is a morphism from the terminal object to GG. The correct morphism from GG to GG here is id.

    AD

    diff, v57, current

  19. Previous edit was not exact, albeit current version does not appear correct either.

    AD

    diff, v57, current

  20. Corrected morphism now, minor detail (see edit thread).

    AD

    diff, v57, current

  21. Corrected morphism now, minor detail (see edit thread).

    AD

    diff, v57, current

  22. Corrected morphism now, minor detail (see edit thread).

    AD

    diff, v57, current

  23. Code error.

    AD

    diff, v57, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTime7 days ago

    Right, thanks. But let’s call the terminal map maybe “pp”, not “gg” (since the latter looks like it should denote a group element).

    diff, v58, current

    • CommentRowNumber25.
    • CommentAuthorRodMcGuire
    • CommentTime7 days ago

    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.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTime6 days ago

    Thanks for the alert. Looking through the page history, the link was broken in revision 54 by John Baez.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTime6 days ago

    Am taking care of the entry now…

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTime6 days ago

    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.

    diff, v59, current

1 to 28 of 28

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