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A number of nLab pages (e.g. ring, associative algebra, graded algebra, differential graded algebra, differential graded-commutative algebra) define (or mention) the notion in question as a monoid in some monoidal category.
Is this possible in the case of divided power algebras?
I’m going by instinct, but it looks to me like the theory of divided power algebras is a plethory, which can be abstractly formulated in the language of monoids, and that divided power algebras themselves will be given by actions of that plethory, similar to the way lambda rings work. I don’t see off-hand how divided power algebras are monoids of some sort themselves, but maybe I’m missing something.
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