David Roberts
topological fundamental bigroupoid

For a locally well-behaved space XX (see below) the fundamental bigroupoid Π 2(X)\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 TopTop, 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).

2-well-connected spaces

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 XX is 2-well-connected if it admits a basis of neighbourhoods UU that are 1-connected and for any basepoint uUu\in U, the induced map π 2(U,u)π 2(X,u)\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 2S^2 embedded in R 3\mathbf{R}^3.

Set-theoretic description of Π 2(X)\Pi_2(X)

  • Objects: these are just points of XX.

  • 1-arrows: paths in XX. The 0-source and 0-target of a path are the evaluations at the endpoints, s 0(γ)=γ(0)s_0(\gamma) = \gamma(0) t 0(γ)=γ(1)t_0(\gamma) = \gamma(1). There is an identity 1-arrow for each object xx which is the constant path at xx, denoted x̲:IX\underline{x}\colon I \to X

  • 2-arrows: Define a surface in a space XX to be a map ff from the square I 2I^2 into XX, such that f(,0)f(-,0) and f(,1)f(-,1) are constant paths (in other words, they are bigons in XX). A homotopy between surfaces f 0f_0 and f 1f_1 is a map F:I 3XF:I^3 \to X such that F(0,,)=f 0F(0,-,-) = f_0 and F(1,,)=f 1F(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][f]. This has well-defined 1-source s 1[f]=f(0,)s_1[f] = f(0,-) and 1-target t 1[f]=f(1,)t_1[f] = f(1,-), which are paths in XX (independent of the representative surface ff), and 0-source s 0[f]=f(0,0)s_0[f] = f(0,0) and 0-target t 0[f]=f(1,1)t_0[f] = f(1,1), which are points of XX.

The set of 2-arrows of Π 2(X)\Pi_2(X) is the set of 2-tracks in XX. There is an identity 2-track for a path γ\gamma, denoted id γid_\gamma, represented by the composite

I 2pr 2IγX. I^2 \stackrel{pr_2}{\to} I \stackrel{\gamma}{\to} X.

Let c:I 2[0,2]×Ic:I^2 \to [0,2]\times I be the isomorphism given by multiplying the first coordinate by 2. Then the composition [f]+[g][f]+[g] of a pair of 2-tracks [f][f] and [g][g] with s 1[f]=t 1[g]s_1[f] = t_1[g] is represented by

f+g:I 2[0,2]×IX f+g: I^2 \to [0,2]\times I \to X

where the second map is the obvious pasting with ff shifted to have domain [1,2][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 Π 2(X)̲ 1\underline{\Pi_2(X)}_1 has as objects the paths in XX, and as arrows the 2-tracks, with composition, identity and inverse as described above.

We have the functors (s 0,t 0):Π 2(X)̲ 1X×X(s_0,t_0):\underline{\Pi_2(X)}_1 \to X \times X and XΠ 2(X)̲ 1X \to \underline{\Pi_2(X)}_1, the latter sending a point to the constant path.

Topological version

We begin with a topological result about path spaces (Andrew - this isn’t necessary for the case when XX is a manifold)

Theorem:(Wada, DMR) When XX is 2-well-connected, the path space X IX^I with the compact-open topology is 1-well-connected.

The basic neighbourhoods of Π 2(X)\Pi_2(X) are as follows. Let [f][f] be a 2-track, with γ 0=s 1[f]\gamma_0 = s_1[f], γ 1=t 1[f]\gamma_1 = t_1[f], x 0=s 0[f]x_0 = s_0[f] and x 1=t 0[f]x_1 = t_0[f]. Let N 0N_0 be a basic neighbourhood of γ 0\gamma_0 and N 1N_1 a basic neighbourhood of γ 1\gamma_1. Also let M 0M_0 be a basic neighbourhood of x 0x_0 and M 1M_1 a basic neighbourhood of x 1x_1, subject to the following conditions:

  • ev 0 1(M 0)N 0N 1ev_0^{-1}(M_0) \subset N_0 \cap N_1, and
  • ev 1 1(M 1)N 0N 1ev_1^{-1}(M_1) \subset N_0 \cap N_1

where ev 0,ev 1ev_0,ev_1 are the evaluation maps X IXX^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 π 2(X,x 0)\pi_2(X,x_0). Am I on the right (2-)track?

DR: Correct on both accounts. It turns out that Π 2(X)X I× X 2X I\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.

Revised on January 20, 2010 23:20:22 by David Roberts (203.24.207.107)