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Stream: learning: questions

Topic: Tensoring categories with monoidal categories

Emily (Oct 22 2021 at 10:18):

Monoids (or really any category with arbitrary coproducts) are tensored over $\mathsf{Sets}$ via $X\odot M\cong\coprod_{x\in X}M$.

IIRC I've seen it claimed before that similarly monoidal categories are tensored over categories; what is the explicit description of this tensor?