Vtyjkyuk

I'm trying to write down a clean definition of the fundamental $2$-groupoid $pi_leq 2(X)$ of a topological space $X$. Specifically, I'm concerned with how to properly define $2$-morphisms. Here is what I have:

The fundamental $2$-groupoid $pi_leq 2(X)$ is the strict $2$-category where

• The objects of $pi_leq 2(X)$ are the points of $X$.


• The $1$-morphisms $xrightarrow y$ of $pi_leq 2(X)$ are the
  continuous maps $f:[0,|f|]rightarrow X$ with $f(0)=x$
  and $f(|f|)=y$. Here $|f|geq 0$.


• For $xxrightarrowfyxrightarrowgz$ define $gcirc f$ as the continuous
  map $[0,|f|+|g|]rightarrow X$ given by
  $$
  (gcirc f)(t)
  =
  begincases
  f(t) & 0leq tleq|f| \
  g(t-|f|) & |f|leq tleq|f|+|g|
  endcases
  $$

Of course, we are deviating from the usual definition of a path as a map $[0,1]rightarrow X$ to make the composition of $1$-morphisms associative on the nose. However, this seems to create a new problem. The $2$-morphisms should be homotopies of paths, but a homotopy between two maps is only defined if the maps have common domain and codomain. To remedy the situation, I've tried the following:

• For $xxrightarrowfy$, let $overlinef:xrightarrow y$ be the continuous map $[0,1]rightarrow X$ given by $overlinef(t)=f(|f|cdot t)$. Then the $2$-morphisms $alpha:fRightarrow g$ in $pi_leq 2(X)$ are the continuous maps $alpha:[0,1]times[0,1]rightarrow X$ with $alpha(t,0)=overlinef(t)$, $alpha(t,1)=overlineg(t)$, $alpha(0,t)=x$, and $alpha(1,t)=y$.

Is this the standard construction? I'm having trouble finding literature on this.

This is a question which raises all sorts of interesting issues!

I think it is easier to look for double groupoids rather than $2$-groupoids, because it is more symmetric with regard to directions and before taking homotopy classes of some kind, the notion of partial composition in a given direction is very clear, and extends to all dimensions. See also this question on mathoverflow.

I though about this for 9 years starting in 1965 with the aim, as it seemed likely that the proof I had written out many times for my topology book extended to get a $2$-dimensional van Kampen type theorem, and got myself quite confused. So it was "an idea for a proof in search of a theorem".

The breakthrough came with Philip Higgins in 1974 when, as suggested by work of J H C Whitehead on second relative homotopy groups, we started looking not at spaces but at pairs of spaces, $(X,A)$. So the natural idea was to look at maps of square $I^2$ into $X$ which took the vertices to the base point and edges into $A$. Then all became easy, not trivial, but easy. The details are in this book, which has a free pdf. A key picture for the definition of compositions being well defined is

The point is that that if $alpha, beta$ are two squares as above then their classes $[alpha], [beta]$ are homotopy classes rel vertices of maps $I^2 to X$ where during the homotopies the boundary of $I^2$ stays in $A$. So $ partial^+_2[alpha] =partial^2_-[beta] $ means there is a homotopy $h$ of two loops in $A$, and such a homotopy is arbitrary. So when you try to prove independence of choices, you get the above figure, which has a big hole in the middle! But the bottom edges of the hole are all constant, so you can fill in the bottom square. Now fill in the hole by a retraction. All the faces of the "hole" are in $A$ so in the retraction the top face is also in $A$. So the whole block gives a homotopy $$alpha +_2 h +_2 beta simeq alpha' +_2 h'+_2 beta'$$ of the type required.

This construction we called $rho_2(X,A,a)$. It contains $pi_2(X,A,a)$ . From this you can get as a substructure a $2$-groupoid, if you really want it.

There are papers dealing with the absolute case for Hausdorff spaces, see K.A. Hardie, K.H. Kamps and R.W. Kieboom, ‘A homotopy 2–groupoid of a Hausdorff space’, Applied Cat. Structures 8 (2000), 209–234, and subsequent papers dealing with double groupoids in a related fashion. But these papers have not it seems led to new calculations of homotopical invariants, as has the above relative construction.

Edit Feb 9, 2014: The above construction of a homotopy double groupoid of a pair is used to prove a $2$-dimensional van Kampen type theorem, one of whose consequences, an excision type theorem for second relative homotopy groups, is explained in my answer to this stackexchange question.