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To what would Prof be analogously equivalent?
To total categories.
Wouldn’t it be presheaf categories, i.e. free cocomplete categories?
Oh, sorry, yes, I was reading much too quickly.
Daniel, I’m sorry, but that’s just wrong. I’m going to have to revert.
Normally I refer to posets with finite joins as a join-semilattice.
in the natural numbers ordered by divisibility, {powers of 2} and {powers of 3} are both downsets with no maximal elements.
in the divisibility lattice, $0$ is the maximal element of everything. If we say it is a power of $2$, $0 = 2^\infty$ there is no downset that just contains powers of $2$.
This strikes me as a rather unsatisfactory fix.
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