Things as categories

Delooping category

The definition of a monoid is wholly equivalent to that of a single-object category.

Given a monoid 𝑀 =(𝑀, ∘), one may construct a category 𝐁𝑀 consisting of a single object ∙ such that 𝖡𝑀( ∙, ∙) =𝑀 and composition is given by ( ∘). Conversely, 𝖢( ∙, ∙) forms a monoid under composition for any ∙ ∈Ob⁡𝖢. Thus a category is the oidification of a monoid, i.e. a monoidoid.

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