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1. Page created. Idempotent monoids should be to monoids as idempotent monads are to monads.

I’ve added the examples of idempotent elements in (ordinary) monoids (1), idempotent morphisms in categories (2), solid rings (3), idempotent monads (4), idempotent $1$-morphisms in bicategories (5), and “solid ring spectra” (6) ―What are other examples?

Also, should idempotent monoids have a unit? The examples 1 and 2 I mentioned above don’t, but 3, 4, and 6 do, while whether 5 does or doesn’t seems to vary a bit among the literature (AFAIU).

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

Nice!

I have added hyperlinks to more of the technical term appearing (such Spectra).

• CommentRowNumber3.
• CommentAuthorvarkor
• CommentTimeSep 19th 2021

Also, should idempotent monoids have a unit?

When there is no unit, surely the appropriate terminology is “idempotent magma”.

2. Thanks, Urs! (I really need to get in the habit of adding more hyperlinks when writing things in the nLab!)

When there is no unit, surely the appropriate terminology is “idempotent magma”.

Ah, true. I had Kerodon in mind, which talks about non-unital monoidal categories, non-unital monoids, non-unital monoidal functors, etc. Going to it again, I see Tag 00BQ reads

“The terminology of Definition 2.1.0.3 is not standard. Most authors use the term semigroup for what we call a nonunital monoid.”

I think probably the page should at least mention the various possible names (including “semigroup” or “semigroup object”, the latter of which is explicitly used in Section 6 of the semigroup page), but I’m not sure which one is the best name among these three.

• CommentRowNumber5.
• CommentAuthorvarkor
• CommentTimeSep 19th 2021

I agree “idempotent semigroup (object)” seems most suitable when associativity is present. I think “object” can be omitted in this instance, as there ought to be no confusion.

3. I was about to change the page, but then I noticed that we would need to rewrite e.g.

(Non-unital) idempotent monoids in $\mathcal{C}$ are the same as (non-unital) strong monoidal functors from the punctual monoidal category $\mathsf{pt}$.

to something like

Idempotent monoids (resp. idempotent semigroups) in $\mathcal{C}$ are the same as strong monoidal functors (resp. ???) from the punctual monoidal category $\mathsf{pt}$.

What should we replace “???” for? I think the natural choice would be “strong semigroupal functor”, but this is a terminology that is very rarely used (I was going to say it wasn’t used at all, but it seems there are a few papers actually using that name!), and its meaning is definitely less clear than “non-unital strong monoidal functor”.

How should we proceed here? Should we change it to “idempotent semigroup” and create pages like “semigroupal category”, “strong semigroupal functor”, etc. or should we just go with “non-unital monoid” and “non-unital strong monoidal functor” after all?

• CommentRowNumber7.
• CommentAuthorvarkor
• CommentTimeSep 20th 2021

It’s reasonable to redirect idempotent semigroup to this page, even if it’s mainly about idempotent monoids. I don’t see a problem with terminology like semigroupal category, considering we already have magmoidal category and the like.

4. Okay! I’ve edited the page to default to “idempotent semigroup”. Thanks, Varkor! :)

5. added the notion of a “strict idempotent monoid” + examples involving Day convolution

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