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1. • CommentRowNumber1.
   • CommentAuthorJ-B Vienney
   • CommentTimeJul 28th 2022
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexA 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. <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/15">v15</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

   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

2. • CommentRowNumber2.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 13th 2022
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexExplained more generaly the construction of the symmetric algebra in a [[CMon-enriched symmetric monoidal category]]. <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/17">v17</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

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

   diff, v17, current

3. • CommentRowNumber3.
   • CommentAuthorJ-B Vienney
   • CommentTimeNov 19th 2022
   • (edited Nov 19th 2022)
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexAdded that permutations $\sigma:A^{\otimes n} \rightarrow A^{\otimes}$ are defined in the entry [[symmetric monoidal category]] <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/21">v21</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

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

   diff, v21, current

4. • CommentRowNumber4.
   • CommentAuthorJ-B Vienney
   • CommentTimeNov 23rd 2022
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexFinally 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. <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/22">v22</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

   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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 26th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 &lbrack;[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/23">v23</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

   added pointer to

   • Sergey Gorchinskiy, Vladimir Guletskii, Symmetric powers in abstract homotopy categories, Adv. Math., 292 (2016) 707-754 [arXiv:0907.0730, doi:10.1016/j.aim.2016.01.011]

   diff, v23, current

6. • CommentRowNumber6.
   • CommentAuthordasistmiregal
   • CommentTimeJul 29th 2023
   • (edited Jul 29th 2023)
   • PermaLink
   Author: dasistmiregal
   Format: MarkdownItexShould $\operatorname{S}^n V$ 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 $0$ and in the vector space case, when $n = 1$, the coequalizer is $0$ and when $n = 2$, the resulting elements are anti-symmetric. Am I missing something? I think it's correct to define it as the coequalizer of the $S_n$-action.

   Should $\operatorname{S}^n V$ 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 $0$ and in the vector space case, when $n = 1$, the coequalizer is $0$ and when $n = 2$, the resulting elements are anti-symmetric. Am I missing something?

   I think it’s correct to define it as the coequalizer of the $S_n$-action.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 29th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the alert. The first "cokernel" here (which originates way back in [revision 1](https://ncatlab.org/nlab/revision/symmetric+algebra/1)) is probably a typo for "image", meaning to refer to the standard construction laid out for instance here: [pdf (p. 27)](https://www.cis.upenn.edu/~cis6100/diffgeom7.pdf#page=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 :-).

   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 :-).

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 29th 2023
   • (edited Jul 29th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexchanged "cokernel" to "image" and added pointer to: * [[Jean Gallier]], *Tensor Algebras, Symmetric Algebras and Exterior Algebras* &lbrack;[pdf](https://www.cis.upenn.edu/~cis6100/diffgeom7.pdf)&rbrack;, section 22 in: *Notes on Differential Geometry and Lie Groups* (2011) * [[Jean Gallier]], [[Jocelyn Quaintance]], *Tensor Algebras and Symmetric Algebras*, Ch 2 in: *Differential Geometry and Lie Groups -- A second course*, Geometry and Computing **13**, Springer (2020) &lbrack;[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/24">v24</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

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

   • Jean Gallier, Tensor Algebras, Symmetric Algebras and Exterior Algebras [pdf], section 22 in: Notes on Differential Geometry and Lie Groups (2011)

   • Jean Gallier, Jocelyn Quaintance, Tensor Algebras and Symmetric Algebras, Ch 2 in: Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [doi:10.1007/978-3-030-46047-1, webpage]

   diff, v24, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Jean-Baptiste Vienney]], *String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\geq 0}$-linear categories* &lbrack;[arXiv:2308.02094](https://arxiv.org/abs/2308.02094)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/25">v25</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

   added pointer to:

   • Jean-Baptiste Vienney, String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\geq 0}$-linear categories [arXiv:2308.02094]

   diff, v25, current

10. • CommentRowNumber10.
    • CommentAuthornLab edit announcer
    • CommentTimeAug 19th 2024
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexAdded [[algebra - contents]] to context sidebar Anonymouse <a href="https://ncatlab.org/nlab/revision/diff/symmetric+algebra/26">diff</a>, <a href="https://ncatlab.org/nlab/revision/symmetric+algebra/26">v26</a>, <a href="https://ncatlab.org/nlab/show/symmetric+algebra">current</a>

    Added algebra - contents to context sidebar

    Anonymouse

    diff, v26, current