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A monoid can be seen as a one-object category.

Is there analogous thing for monoids in a monoidal category $(M, \otimes, I)$? Can I form some kind of one-object category from a monoid in $M$?

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2 Answers 2

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It is a one-object category enriched over $M$, see http://ncatlab.org/nlab/show/enriched+category#InMonoidCat.

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    $\begingroup$ And then a standard way of getting an ordinary (one-object) category from this is by applying the change-of-base functor $\hom(I, -): M \to \mathbf{Set}$. This takes enriched monoids to ordinary monoids. $\endgroup$
    – user43208
    Commented Nov 4, 2013 at 17:45
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A monoidal category is a one object 2-category

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