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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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when is an integer sequence the trace of a monad on FinSet?

Given $(a_n \in \mathbb{N})$, when is there a monad $T$ on $\mathrm{FinSet}$ such that $$ | T(n) | = a_n\quad\forall n\in \mathbb{N}\:? $$
Felix Dilke's user avatar