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    • CommentRowNumber1.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2022

    A few words and an hyperlink to a page where I will put my conjectural characterization of symmetric powers in symmetric monoidal categories enriched over modules over a +\mathbb{Q}^{+}-algebra.

    diff, v15, current

    • CommentRowNumber2.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 13th 2022

    Explained more generaly the construction of the symmetric algebra in a CMon-enriched symmetric monoidal category.

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 19th 2022
    • (edited Nov 19th 2022)

    Added that permutations σ:A nA \sigma:A^{\otimes n} \rightarrow A^{\otimes} are defined in the entry symmetric monoidal category

    diff, v21, current

    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 23rd 2022

    Finally almost succeeded to prove this: symmetric powers in a symmetric monoidal +\mathbb{Q}^{+}-linear category are characterized among the countable families of objects as forming a special connected graded quasi-bialgebra. Hope to add the reference soon.

    diff, v22, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023

    added pointer to

    diff, v23, current

    • CommentRowNumber6.
    • CommentAuthordasistmiregal
    • CommentTimeJul 29th 2023
    • (edited Jul 29th 2023)

    Should S nV\operatorname{S}^n V be defined as the cokernel of the symmetrisation map? That’s quite a strange definition which I don’t think it correct. Even in char 00 and in the vector space case, when n=1n = 1, the coequalizer is 00 and when n=2n = 2, the resulting elements are anti-symmetric. Am I missing something?

    I think it’s correct to define it as the coequalizer of the S nS_n-action.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2023

    Thanks for the alert. The first “cokernel” here (which originates way back in revision 1) is probably a typo for “image”, meaning to refer to the standard construction laid out for instance here: pdf (p. 27).

    I’ll make a quick fix to the entry now, but won’t be doing it justice (since I am still on family vacation and not allowed to be online :-).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2023
    • (edited Jul 29th 2023)

    changed “cokernel” to “image” and added pointer to:

    diff, v24, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2023

    added pointer to:

    diff, v25, current