# Reference for the delooping of groups (or rings)

This nlab article defines the delooping groupoid of a group, but it does not provide a reference. I am using deloopings of rings, and I would like to refernce to a more classical source.

Could you please recommend a reference for the delooping functor? Btw, where were deloopings of groups (or rings) first considered?

• To be honest: I always found it strange to distinguish between a group $G$ and the corresponding one-object groupoid. The data are equivalent, and I have never seen a situation where this is really necessary (does someone know such a situation?). Similarly, it is not really important to distinguish between a poset and the induced thin category, but this seems to be commonly accepted and there is no extra notation. And even more confusingly, $BG$ also denotes (more often, actually) the classifying space of $G$. – HeinrichD Nov 8 '16 at 19:37
• @HeinrichD - The category of groups is not equivalent to the 2-category of one-object groupoids. It follows, for instance, that a "sheaf of groups" is not the same as "a (2-)sheaf of one-object groupoids" -- the latter are also called "gerbes", and not every gerbe is globally the delooping of a group. – Mike Shulman Nov 8 '16 at 22:28
• @MikeShulman: There is no category which is equivalent to a 2-category, because equivalences are only defined between objects of the same type. The 2-category of groups is equal to the 2-category of one-object groupoids. For the same reason, it is clear that sheaves do not coincide with 2-sheaves, regardless of their values. – HeinrichD Nov 8 '16 at 22:49
• @HeinrichD It's standard to identify a category with the 2-category obtained from it by adding only identity 2-cells. But the point is the same whether you phrase it as "the category of groups is not equivalent to the 2-category of one-object groupoids" or "groups form a 1-category whereas one-object groupoids form a 2-category". – Mike Shulman Nov 9 '16 at 4:07
• Of course, you can define a 2-category whose objects "are" groups that is equivalent to the 2-category of one-object groupoids, just as you can define a category whose objects "are" groups that is equivalent to the category of sets. But that sort of chicanery robs definitions of all meaning. The natural categorical structure formed by groups is a 1-category, whereas the natural categorical structure formed by one-object groupoids is a 2-category, and the two are not the same in any sense. – Mike Shulman Nov 9 '16 at 4:10