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nLab > Latest Changes: magmoidal category

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    Anonymous

    v1, current

  2. The categorification of a magma

    Anonymous

    v1, current

    • CommentRowNumber3.
    • CommentAuthorvarkor
    • CommentTimeMay 5th 2021

    With reference to the terminology “monoidal category”, I think this ought to be called a “magmoidal category” (this appears to have been used more frequently in practice too).

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 6th 2021

    Changed name as suggested. I agree, it goes better with monoidal category, which is the categorification of a monoid, and appears in the literature: Davydov’s Nuclei of categories with tensor products and Lack and Street’s Triangulations, orientals, and skew monoidal categories, and honourable mention to H. Eades III in Introducing a New Project on The Combination of SubstructuralLogics and Dependent Type Theory (extended abstract).

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 6th 2021

    Also added example of “…with unit”, following Eades. This includes incoherent unitors, since there is no associator to be compatible with. However, I do wonder if one should ask that ρ I=λ I\rho_I = \lambda_I…

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 7th 2021

    Added example of Ackermann groupoids.

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeMay 7th 2021

    However, I do wonder if one should ask that ρ I=λ I\rho_I = \lambda_I…

    I agree that one should: as a term rewriting system, these rewrites should be convergent, so any choice of path should be equal.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 12th 2021

    Added note on a potentially necessary weak coherence condition for magmoidal categories with units, namely that there should be a unique coherence iso I⊗I→II\otimes I \to I, so the unitors should coincide. Also noted the usual unitor coherence condition can’t even be stated in the absence of associators.

    diff, v5, current

  9. Added reference

    Anonymouse

    diff, v8, current

  10. Added the fact that every magma is a discrete magmoidal category

    Anonymouse

    diff, v8, current

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeFeb 4th 2024

    Added “magmal category” terminology. This seems more reasonable than “magmoidal”, since it is inspired by the term “magma”, which does not contain the “-oid” suffix, in contrast to “monoid” for “monoidal category”.

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2024

    Maybe magmatic?!

    • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeFeb 4th 2024

    True, that’s yet another possibility… (and one it seems Bartosz Milewski has employed in this article).

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