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]

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 :-).

]]>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.

]]>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]

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.

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

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

]]>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.

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