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The Drinfeld center construction is intended to be a categorification of the center of a monoid. It seems to be folklore (eg this answer or this one) that when the Drinfeld center is taken over a monoid $M$ (viewed as a category $BM$ with one single object) the usual notion of the center of the monoid is recovered.

If we denote $\bullet$ to this unique object (so that $\mathrm{End}_{BM}(\bullet) = \mathrm{Hom}_{BM}(\bullet, \bullet) =M$), then we are forced to define $\bullet \otimes \bullet := \bullet$ and of course $\bullet$ must be the unit for the monoidal structure. However, a foundational property of monoidal categories is that for any such $(\mathcal{C}, \otimes, 1)$,we have that $\mathrm{End}_{\mathcal{C}}(1)$ is a commutative monoid (eg Lemma 1.1 of Turaev, Vladimir; Virelizier, Alexis, Monoidal categories and topological field theory.)

Essentially this says that $BM$ cannot be monoidal unless $M$ is a commutative monoid. Then

  1. Am I making a blunder or that interpretation is simply not true?
  2. What is the correct statement that recovers the center of a monoid from the Drinfeld construction?
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    $\begingroup$ I think what we want is rather to see $M$ as a monoidal category whose objects are elements of $M$ and morphisms are identities. $\endgroup$
    – Adrien
    Commented Mar 6, 2021 at 13:33
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    $\begingroup$ The answer by AAK you point out is a bit misleading: there is a notion of center for an arbitrary category, the endomorphisms of the identity functor, which for $BM$ recover its classical center. Those two notions are closely related though. $\endgroup$
    – Adrien
    Commented Mar 6, 2021 at 13:40

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