Monoids as categories

The definition of a monoid is wholly equivalent to that of a single-object category. Given a monoid $𝑀 = (𝑀, \circ)$ , one may construct a category $𝐁 𝑀$ consisting of a single object $∙$ such that $𝐁 𝑀 (∙, ∙) = 𝑀$ and composition is given by $(\circ)$ . Conversely, $𝖢 (∙, ∙)$ forms a monoid under composition for any $∙ \in O b 𝖢$ . Thus a category is the oidification of a monoid, i.e. a monoidoid.

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