For a locally well-behaved space $X$ (see below) the fundamental bigroupoid $\Pi_2(X)$ as defined in the PhD thesis of Danny Stevenson and in the 2001 article by Hardie-Kamps-Kieboom can be internalised in $Top$, that is, the sets of objects, 1- and 2-arrows can be given topologies such that all the maps involved in the definition of the bigroupoid are continuous.
This page will explain the construction (soon).
The existence of topological fundamental groupoid is tied up with the existence of a universal covering space. This requires local conditions on the topological space at hand involving fundamental groups of neighbourhoods. For the topological fundamental bigroupoid, we also need local conditions, this time on first and second homotopy groups.
Definition: A topological space $X$ is 2-well-connected if it admits a basis of neighbourhoods $U$ that are 1-connected and for any basepoint $u\in U$, the induced map $\pi_2(U,u) \to \pi_2(X,u)$ is the zero map.
This generalises 1-well-connected, which is the condition necessary for the existence of a universal covering space (in existing terminology, locally path-connected and semilocally simply-connected).
For example, any CW complex or manifold is 2-well-connected. A space that is not 2-well-connected is the analogue of the Hawaiian earring, built from spheres $S^2$ embedded in $\mathbf{R}^3$.
Objects: these are just points of $X$.
1-arrows: paths in $X$. The 0-source and 0-target of a path are the evaluations at the endpoints, $s_0(\gamma) = \gamma(0)$ $t_0(\gamma) = \gamma(1)$. There is an identity 1-arrow for each object $x$ which is the constant path at $x$, denoted $\underline{x}\colon I \to X$
2-arrows: Define a surface in a space $X$ to be a map $f$ from the square $I^2$ into $X$, such that $f(-,0)$ and $f(-,1)$ are constant paths (in other words, they are bigons in $X$). A homotopy between surfaces $f_0$ and $f_1$ is a map $F:I^3 \to X$ such that $F(0,-,-) = f_0$ and $F(1,-,-) = f_1$ and.. (conditions expressing it is a relative homotopy)
Definition: A 2-track is a homotopy class of surfaces, and will be written $[f]$. This has well-defined 1-source $s_1[f] = f(0,-)$ and 1-target $t_1[f] = f(1,-)$, which are paths in $X$ (independent of the representative surface $f$), and 0-source $s_0[f] = f(0,0)$ and 0-target $t_0[f] = f(1,1)$, which are points of $X$.
The set of 2-arrows of $\Pi_2(X)$ is the set of 2-tracks in $X$. There is an identity 2-track for a path $\gamma$, denoted $id_\gamma$, represented by the composite
Let $c:I^2 \to [0,2]\times I$ be the isomorphism given by multiplying the first coordinate by 2. Then the composition $[f]+[g]$ of a pair of 2-tracks $[f]$ and $[g]$ with $s_1[f] = t_1[g]$ is represented by
where the second map is the obvious pasting with $f$ shifted to have domain $[1,2]$. This is independent of representative as one would imagine, and the identity 2-track is an identity for this composition.
AS: Can you add in the structure maps here and the properties you want. Or link to the appropriate definition of bi/2 category in the nLab so that I can work them out! (I’m pretty sure I know what the structure maps should morally be, so it’s the properties that I’m most interested in since those may or may not mean that the moral maps need minor alteration to fit the desired properties.) I’m particularly interested in composition of 1-arrows.
DR: I’ll define a bigroupoid a little idiosyncratically, as it helps show the structure we are dealing with is internal to the appropriate category.
Definition: The groupoid of 2-tracks $\underline{\Pi_2(X)}_1$ has as objects the paths in $X$, and as arrows the 2-tracks, with composition, identity and inverse as described above.
We have the functors $(s_0,t_0):\underline{\Pi_2(X)}_1 \to X \times X$ and $X \to \underline{\Pi_2(X)}_1$, the latter sending a point to the constant path.
We begin with a topological result about path spaces (Andrew - this isn’t necessary for the case when $X$ is a manifold)
Theorem:(Wada, DMR) When $X$ is 2-well-connected, the path space $X^I$ with the compact-open topology is 1-well-connected.
The basic neighbourhoods of $\Pi_2(X)$ are as follows. Let $[f]$ be a 2-track, with $\gamma_0 = s_1[f]$, $\gamma_1 = t_1[f]$, $x_0 = s_0[f]$ and $x_1 = t_0[f]$. Let $N_0$ be a basic neighbourhood of $\gamma_0$ and $N_1$ a basic neighbourhood of $\gamma_1$. Also let $M_0$ be a basic neighbourhood of $x_0$ and $M_1$ a basic neighbourhood of $x_1$, subject to the following conditions:
where $ev_0,ev_1$ are the evaluation maps $X^I \to X$ at 0 and 1 resp. (Andrew: For ‘basic neighbourhoods’ read ‘charts’ in the manifold case)
To be continued…
Andrew: Quick question to check that I’m following the idea. If I fix two 1-arrows (with suitable endpoints) and look at the 2-tracks between them, will that end up with the discrete topology? This seems to fit with the fact that “covering spaces” keep getting mentioned! In particular, if the 1-arrows are both the identity at some object, then I should just end up with $\pi_2(X,x_0)$. Am I on the right (2-)track?
DR: Correct on both accounts. It turns out that $\Pi_2(X) \to X^I \times_{X^2} X^I$ is a covering space. A given fibre of this map is the set (=discrete space) of 2-tracks between 2 paths. (I should also point out that I allow covering spaces to have empty fibres, which is important when the base space is not connected.) Indeed the construction of this covering space in our category of manifolds is one of the central challenges of this exercise.
AS: Good. Then I’m along the right lines. The fact that you were taking quotients worried me, as that’s dangerous for staying in manifolds, but as your quotients end up fibrewise discrete then I’m happy again.