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If groupoids are "indexed groups", wouldn't that same naming scheme imply that categories should be called "monoidoids", or more sensibly, why aren't categories called "monoids" and monoids called... "mon"s? "mono"s?

Cactus
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    Mathematical terminology is seldom particularly logical. Are you asking about the historical origins of this terminology? – Eric Wofsey Jul 18 '19 at 02:24
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    @EricWofsey I guess a more causally sequentialized version of this question would be, why the "groupoid" name is used in this way, inconsistently with "monoid". – Cactus Jul 18 '19 at 02:44
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    Actually, does the name "monoid" even make sense before the category theoretical formulation? What is "one" ("mono") in a monoid in its algebraic formulation? – Cactus Jul 18 '19 at 02:45
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    I agree with you that this would make more sense and match more closely with the fact that categories are basically "multi-sorted monoids", and be more consistent with what happens for groups and groupoids, which sounds much nicer. But the convention everyone got used to are not these ones. Probably if we were to name these notions now we would chose something in the flavour that you suggested. At the time they were named though, people did not have the same understanding of it as we do now – Thibaut Benjamin Jul 18 '19 at 08:39
  • @ThibautBenjamin: so what about monoids is mono in a non-categorical understanding? What I mean is, in a CT understanding of a monoid, there is a single object, so mono's meaning of only, unique readily applies, but the name comes from a simple algebraic formulation, and I'm not seeing anything mono in that. – Cactus Jul 18 '19 at 09:04
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    I do not know where the "mono" part comes from in monoid. But you can check the answers here https://math.stackexchange.com/questions/156952/why-the-terminology-monoid/167900 – Thibaut Benjamin Jul 18 '19 at 09:23
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    A monoid has an identity; if there is an identity element, then it is unique. https://en.wiktionary.org/wiki/mono- – fosco Jul 18 '19 at 09:45
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    It's perhaps worth noting that groupoids were invented (or at least, named) before categories, and monoids slightly after. – Arnaud D. Jul 18 '19 at 12:09

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