The String Coffee Table

Ieke Moerdijk is giving a series of talks in Hamburg on the general topic of orbifolds.

The first lecture today was mainly concerned with highlighting the right way to think about orbifolds, that which allows to neatly talk about maps between them and about extra structure on them.

If I may rephrase this point of view in my words, I would state it in terms of a slogan as

Orbifolds are to be thought of as decategorified groupoids.

Apart from technical issues related to the fact that one wants everything to be smooth in a suitable sense, the simple (and well known) idea is the following.

To every topological space $M$ on which some group $G$ acts (or rather, on which several such groups act locally) , we may associate the action groupoid

whose objects are the points of $x$ and which has a morphism between $x$ and $y$ if and only if there is a $g\in G$ such that acting with $g$ on $x$ yields $y$.

To divide out by the action of $G$, i.e. to identify points on a commong $G$-orbit amounts to nothing but passing to the isomorphism classes $\mid G_{M,G}\mid$ of the groupoid $G_{M,G}$.

Passing to isomorphism classes is known as ‘decategorification’. Hence, up to some technical fine print, orbifolds are decategorified groupoids.

As always when some decategorified structure is encountered, one obtains a deeper understanding of its nature by undoing the decategorification and studying the original category it came from. And that’s what Ieke Moerdijk is emphasizing is the right point of view also in the case of orbifolds.

I plan to report on the details of his talks as soon as possible.