Example 8.1.2.1.2 (00WN): Monoids as One-Object Categories

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We have an isomorphism of categories ^[1]

via the delooping functor $\mathsf{B}\colon \mathsf{Mon}\to \mathsf{Cats}$ of

, exhibiting monoids as exactly those categories having a single object.

Proof of Example 8.1.2.1.2.

Omitted.

Footnotes

[1] This can be enhanced to an isomorphism of $2$-categories

between the discrete $2$-category $\mathsf{Mon}_{\mathsf{2disc}}$ on $\mathsf{Mon}$ and the $2$-category of pointed categories with one object.

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