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Atrium > Mathematics, Physics & Philosophy: Brown-Spencer theorem

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    • CommentRowNumber1.
    • CommentAuthorelif
    • CommentTimeFeb 2nd 2016
    Is there any proof of Brown-Spencer Theorem ,which states that the categories of crossed modules resp. categorical groups are equivalent, algebraic version?
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 2nd 2016

    (Ha, when I first saw the title of the thread, my immediate thought was instead of Spencer-Brown, author of Laws of Form. (-: )

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 2nd 2016

    Can you ask that again, elif? I thought the proof was algebraic, in the sense that it doesn’t need homotopy theory or anything.

    • CommentRowNumber4.
    • CommentAuthorelif
    • CommentTimeFeb 2nd 2016
    Is there any equivalance between the category of crossed modules of algebras and categorical groups algebraic correspondence?
    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2016

    There is an equivalence between the category of crossed modules in a semiabelian category CC, and the category of internal groupoids in CC, if that helps. See eg this paper by Janelidze (pdf link).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2016
    • (edited Feb 3rd 2016)

    David, could you put that remark and that link on the relevant nnLab page? Thanks!

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2016

    I was surprised it wan’t there already! Will do.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2016

    Thanks. Also that Brown-Spencer reference is somwhat hidden in the middle of the article here. Maybe when you get to it you could produce one coherent paragraph that points to both references.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2016

    OK, now at crossed module.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2016
    • (edited Feb 3rd 2016)

    Will have to get back to #8. Need to work on some lectures for tomorrow.

    EDIT: also, for reference, there is mention of Brown-Spencer at the page cat-1-group.

    • CommentRowNumber11.
    • CommentAuthorelif
    • CommentTimeFeb 3rd 2016
    What is the internal groupoids in the category of algebras without units?
    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 4th 2016

    Internal category. Up to you, I suppose.

1 to 12 of 12

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