The String Coffee Table

Paths in Categories

Introduction

As argued in particular in [1], orbifolds are best thought of as decategorified groupoids. A point in the orbifold hence appears as an isomorphism class of objects in some category.

Motivated by parallel transport along paths in orbifolds as well as by the study of strings propagating on orbifolds, one would like to similarly understand paths and loops in orbifolds in terms of the representing groupoids.

In the context of what is being called orbifold string topology [2] Lupercio and Uribe had introduced [3] a certain notion of a loop space of a groupoid $G$, called the loop groupoid of $G$.

Their approach rests on the strategy to regard the circle $S^1$ as a groupoid itself in a suitable sense and define the loop space of $G$ as the category of (smooth) functors from $S^1$ to $G$.

Heuristically, a loop in $G$ defined this way is a an alternating concatenation of smooth paths in the object space of $G$ formally composed with morphisms in $G$.

The purpose of the following notes is to indicate that this concept admits also a 2-functorial perspective, which provides a nice way to describe higher order equivariant structures on orbifolds (like (nonabelian) gerbes with connection) in terms of transport 2-functors.

In general, given any smooth category $S$ (groupoid or not), there are generally two different ways to “move” from $a$ to $b$ inside of $S$, where $a$ and $b$ are objects of $S$.

First, there might be a morphism $a \to b$ in $\mathrm{Mor}(S)$. But second, since $S$ is smooth, there might be a smooth path running through the space of objects of $S$, from $a$ to $b$.

We formalize this and introduce the general concept of a category of paths inside a smooth category $S$, whose objects are those of $S$ and whose morphisms are formal composites of smooth paths in the object space of $S$ with morphisms of $S$, subject to certain compatibility relations.

In fact, the main point is that this concept easily categorfies. Given any smooth 2-category $S$ (2-groupoid or not) we can consider the 2-category of 2-paths inside $S$. This has 2-morphisms being formal compositions of smooth surface elements in the object space of $S$ with 2-morphsism of $S$.

We demonstrate that for $S$ representing an orbifold, this concept refines the loop groupoid given by Lupercio and Uribe in that it suspends a 1-category of loops and cobordisms to a 2-category of points, paths and cobordisms. (See the introduction of [4] for why this is desirable.)

Moreover, 2-paths in 2-categories as defined here generalize the definition by Lupercio and Uribe in that it admits cobordisms between paths that are true surfaces, not just “jumps” between orbifold sectors.

We claim that 2-functors from a 2-category of paths inside a groupoid represent equivariant gerbes with connection and parallel surface transport on orbifolds. This applies to abelian bundle gerbes [5] just as well as to nonabelian bundle gerbes [6]. The main concepts are described below. Details of this construction however will be discussed elsewhere.

A special case for this has already been discussed at length. Choosing a good covering of any space gives rise to the Čech-groupoid associated to that space. Regarding this groupoid as an orbifold (it is in fact the embedding groupoid of the trivial orbifold, as defined in section 3.5 of [1]), the cocycle conditions for a locally trivialized 1- or 2-bundle over this space are nothing but the equivariance conditions with respect to this “orbifold” [7, 8].

For this special case the 1- and 2-path categories of paths inside the Čech groupoid have already been studied in section 12.1 of [8]. The following definitions are a straightforward generalization of this concept to arbitary smooth categories. The reader interested in more technical details should hence consult section 12 of [8].