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I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object groupoid $\mathbf{B}G$, which is supposed to be its delooping.

In the case of a ring $R$, we can think of it is as an $\mathsf{Ab}$-enriched category in a corresponding way: i.e. a ringoid with one object. Is it correct to say that this ringoid is the delooping of $R$, and denote it by $\mathbf{B}R$?

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Yes, that's right. This perspective leads to the idea that linear categories are a "many-object" generalization of rings, and naturally occurs in Morita theory.

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  • $\begingroup$ Thanks! I looked up Morita theory - looks interesting. $\endgroup$ – ಠ_ಠ Jun 27 '16 at 5:30

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