The n-Category Café

Let me put it this way:

An orbifold is conveniently thought of # as a groupoid, where isomorphisms connect points that are to be identified under the local group action.

A path in an orbifold is hence a path in a groupoid: a free combination of ordinary paths with morphisms of the groupoid.

This idea plays a prominent role in the field of orbifold string topology #.

Before learning about it’s use there, I had grown fond of the concept of paths in categories - in the sense of free combinations of ordinary paths in the space of objects combined with morphisms in the category - as a convenient means to express $n$-transport on $X$ in terms of local data: here it is the 2-groupoid

associated to the morphism of domains

inside of which we wish to consider 2-paths.

I made some remarks on how to conceive this and the construction used in orbifold string topology in a decent categorical language in these old notes:

$\;\;$ Paths in Categories

If you don’t know what I have in mind when forming the category of $n$-paths inside a smooth $n$-category - how one freely generates new morphisms by combining paths with existing morphisms and dividing out certain relations - you can find the relevant diagrams in these notes.

But at the time of writing these notes I didn’t realize the nice universal structure behind this concept. This I now try to discuss in

$\;\;$ Universal Transition of Transport

I think one obtains the 2-category

of 2-paths inside $Y^\bullet$ as the universal ($P_2(Y) \stackrel{p}{\to} P_2(X)$)-local transition.

(I am freely making use of the terminology of TraTriTra.)

In other words, every $p$-local transition of 2-transport uniquely factors through $P_2(Y^\bullet)$ in a suitable sense.

As a nice corollary of this, we find that the factorization morphism

is the “descending” local trivialized morphism.

To appreciate this, consider the example of $p$-local $i$-trivializations, with $i$ being the obvious injection

Such a $p$-local $i$-trivialization is a line bundle gerbe #.

The factorization morphism $(\mathrm{tra}_y,g,f) : P_2(Y^\bullet) \to T$ associated to this line bundle gerbe is in general not $i$-trivial - reflecting the fact that a line bundle gerbe is a collection of transition bundles instead of transition functions.

But for sufficiently well-behaved $p$ (for instance for any good covering of $X$) $(\mathrm{tra}_y,g,f)$ is in fact $i$-trivializable. Performing this trivialization of the factorization morphism of the original trivialization amounts to passing from the bundle gerbe to its Deligne 3-cocycle #.