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1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexPage created, but author did not leave any comments. Anonymous <a href="https://ncatlab.org/nlab/revision/super+commutative+monoid/1">v1</a>, <a href="https://ncatlab.org/nlab/show/super+commutative+monoid">current</a>

   Page created, but author did not leave any comments.

   Anonymous

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
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   Author: Urs
   Format: MarkdownItexThis does not look right: A "super commutative monoid" must involve a sign when two odd elements are commuted. Even if it's just a graded monoid, then the degrees must add under the monoid operation, so that an even with an odd element gives an odd element.

   This does not look right: A “super commutative monoid” must involve a sign when two odd elements are commuted. Even if it’s just a graded monoid, then the degrees must add under the monoid operation, so that an even with an odd element gives an odd element.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo if "super commutative monoid" here is meant as a more primitive notion of "super vector space", then for the terminology to be justified the entry must consider some kind of non-trivially symmetric braided tensor product on these gadgets. Compare to super vector spaces: These are indeed just $\mathbb{Z}/2$-graded vector spaces in themselves, but get to be called "super" IFF regarded as objects in the non-trivial symmetric monoidal category structure on $\mathbb{Z}/2$-graded vector spaces. If that non-trivial symmetric braiding is not invoked, then $\mathbb{Z}/2$-graded vector spaces are just $\mathbb{Z}/2$-graded vector spaces and not super vector spaces. The same holds for their underlying additive monoids.

   So if “super commutative monoid” here is meant as a more primitive notion of “super vector space”, then for the terminology to be justified the entry must consider some kind of non-trivially symmetric braided tensor product on these gadgets.

   Compare to super vector spaces: These are indeed just $\mathbb{Z}/2$-graded vector spaces in themselves, but get to be called “super” IFF regarded as objects in the non-trivial symmetric monoidal category structure on $\mathbb{Z}/2$-graded vector spaces. If that non-trivial symmetric braiding is not invoked, then $\mathbb{Z}/2$-graded vector spaces are just $\mathbb{Z}/2$-graded vector spaces and not super vector spaces. The same holds for their underlying additive monoids.