Groupoids

My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

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Morita equivalence and the bicategory of bimodules

In the previous post we learned that it is possible to recover the center of a ring from its category of left modules (as an -enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings are said to be Morita equivalent if the categories are equivalent as -enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.

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Centers, 2-categories, and the Eckmann-Hilton argument

The center of a group is an interesting construction: it associates to every group an abelian group in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor (unlike the abelianization ). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category we may associate to an object its automorphism group or endomorphism monoid ), and this is a canonical construction which again doesn’t extend in an obvious way to a functor or . (It merely reflects some special part of the bifunctor .)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group , but of the identity functor , where is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

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