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Qiaochu Yuan
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The classifying space functor may be a monoidal functor out of $\text{Grp}$, but nonabelian groups aren't group objects in $\text{Grp}$. (The group objects in $\text{Grp}$ are precisely the abelian groups, again by the Eckmann-Hilton argument.)

Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741