I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology.

The morphisms in a category with one object have the structure of a monoid. Similarly, a 2-category with one object is a monoidal (1-)category, and so every monoidal category can be thusly viewed as a 2-category with one object by promoting the 1-morphisms to 2-morphisms and the objects to 1-morphisms, just as every monoid $M$ can be viewed as a 1-category $\underline{M}$ with one object by promoting the elements to morphisms.

I have inferred from some context that this process may be called looping/delooping, though I haven't found a reference for this, and the nLab article says stuff which I don't understand. But it's easy enough to see that if $C$ is a category, then the morphisms of $C$ are the objects of $C^I$, where $I=\[0\to1\]$ is the interval category, and this is analogous to the loop space construction in topology $\Omega X=[S^1,X]$. So by demoting morphisms to objects, you get the loop space. In topology for any group we also have $BG$ which satisfies $\Omega BG=G$, hence is called the delooping of $G$; the space whose loop space is $G$. Continuing the analogy, taking a category and promoting its objects to 1-morphisms and 1-morphisms to 2-morphisms, this process may be called "delooping".

For any group $G$, in addition to the delooping $\underline{G}$ category with one object, we also have the "translation category" $TG$ with one object for each group element and exactly one morphism for each pair. $TG$ is a monoidal category, so call its delooping $\overline{G}.$ Like $\underline{G},$ it has one object, and one morphism for every group element. But $\overline{G}$ also has 2-morphisms, exactly one for every pair of 1-morphisms, since it is the delooping of a 1-category, whereas $\underline{G}$ was the delooping of a discrete category (0-category).

So we might consider $G$ the delooping of $TG$ in the category theoretic sense of the word, then we apply the geometric realization of the nerve to both categories, and we get $BG$ and $EG$, the former being the topological delooping.

Is there some sense in which $EG$ is the looping of $BG$, to finish the analogy? It seems not, since $\Omega BG=G$, whereas $EG$ is contractible. Is my definition of looping and delooping of categories correct? What is the relationship between the delooping of a category and the delooping of a space? What is a good reference for understanding this stuff?

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    $\begingroup$ EG is not the looping of BG. Your TG is a 2-group, and so naturally a 2-groupoid with one object. This 2-groupoid is the delooping of TG, and TG is the looping of this 2-groupoid. Just as G sits inside TG, the delooping of G (the one-object groupoid) sits inside the delooping of TG. The quotient of TG by G is the delooping of G, and the same story for the other inclusion. If you suspect there is some iteration going on, you are not wrong. May I suggest arxiv.org/abs/0708.1741 and nyjm.albany.edu/j/2013/19-5.html for some related results. $\endgroup$ – David Roberts Apr 24 '13 at 1:08

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