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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).
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).
Added example of Ackermann groupoids.
However, I do wonder if one should ask that $\rho_I = \lambda_I$…
I agree that one should: as a term rewriting system, these rewrites should be convergent, so any choice of path should be equal.
Added note on a potentially necessary weak coherence condition for magmoidal categories with units, namely that there should be a unique coherence iso $I\otimes I \to I$, so the unitors should coincide. Also noted the usual unitor coherence condition can’t even be stated in the absence of associators.
Added reference
Anonymouse
Maybe magmatic?!
True, that’s yet another possibility… (and one it seems Bartosz Milewski has employed in this article).
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