I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be able to do a similar construction in higher dimensions, and so I was wondering whether a principle similar to the following has been explored. Alas, the idea of these higher groupoids being a category fails for obvious reasons. We'll call this thing a groupesque. I won't give a definition of a groupesque in general - one of my questions is whether such a definition already exists - but I will describe the structure of the "homotopy groupesque." (The name is very conscious - the description that follows is grotesque.)
Throughout, consider all $n$-spheres, including $n$-spheres which are the boundary of $(n+1)$-disks, as being based spaces with a canonically chosen basepoint. Also, I may forget to apply the prefix "homotopy class of" to the word "map"; I apologize, and clarify ahead of time that every map in this post is actually considered up to homotopy.
The setup for the problem is quite long, but I imagine that a reader with an answer won't need to read much of it before they know exactly what I'm asking.
In the fundamental groupoid, our objects are points and maps are homotopy classes paths, maps $I^1\to X$ with appropriate boundary. In particular, our hom-sets are associated, as in any category, to pairs (a,b) of points. Equivalently, they are associated to maps $S^0=\partial I^1\to X$. This latter description has the convenient property that $\hom(a,b)$ is the set of homotopy classes of maps, relative the boundary, which "restrict to (a,b)" in the sense just described.
In the homotopy groupesques $\Pi_n$, we'd like our morphisms to be classes of maps $D^n\to X$ relative the boundary, and we'd like hom-sets to be associated to homotopy classes of maps $S^{n-1}\to X$, so that $\hom(f)$ is the set classes of maps $D^n\to X$ with $f$ as their boundary map (up to homotopy).
The "automorphism groups" of this groupesque in the traditional sense should end up being the hom-groups of constant map; this is nice, since these maps factor through maps from spheres.
But we still have no concept of composition, so when should there be a composition map on $\hom(f)\times\hom(g)$, and which hom-group does it land in? Note that the maps $f:S^{n-1}\to X$ that we're associating hom-sets to are actually elements of some automorphism group of $\Pi_{n-1}$, the previous homotopy groupesque. Basically, we'll say that we we can compose maps at $f$ with maps at $g$ is they lie in the same automorphism group, and hence if we can compose $f$ and $f$ as maps in $\Pi_{n-1}$. (This is where the basedness of all our spheres comes into play. We can compose them if and only if they map the baspoint to the same point.) In particular, composition runs $\hom(f)\times\hom(g)\to\hom(f\star g)$.
Note that this doesn't correspond in any obvious way with the case $n=1$: composition of those maps has nothing to do with a basepoint. This is an acceptable chimera.
Inductively, we can now check that the "automorphism groups" are genuinely what I just claimed they are, but this is where it gets interesting: there is always a composition $\hom(f)\times\hom(f)$ for any $f$, but if it happens to land in $\hom(f)$, then this implies that $f\star f=f$, which is if and only if $f$ is homotopically trivial.
Note that since maps no longer have a "source and target," it is not clear what an inverse is at all in general. However, when $f$ is homotopically trivial and we have a monoid structure this is clear. In that case, we can see that inverses to exist, and that our group structure corresponds to that of the higher homotopy group.
One last note: be sure that you see that $\Pi_1\neq\Pi$! The first homotopy groupesque is not the fundamental groupoid! In particular, hom-sets are associated to homotopy classes of maps $S^0\to X$, so we basically ended up with the skeleton of $\Pi$.
Finally, the questions:
- Does this even make sense? There are so many places I could've gone astray while formulating this in the first place, that it's possible that something went wrong and these "groupesques" do not encode the homotopy groups at all.
- Is this a higher category? I've thought and thought and I just can't seem to get it this object into that framework. The major issue is that even with an $\infty$-category structure, $n$-morphisms are attached to a pair of $(n-1)$-morphisms. It's not clear where "pairs" of anything comes into the picture here. (The higher-category approach seems to give "groups of maps from the $n$-torus" maybe, but not from the sphere.)
- Do groupesques as a generalization of groupoids have a name? Is there a broader generalization of categories where hom-sets are associated to a single thing (rather than two), and some rule describes when a composition is defined and where it lands? (A traditional category is then one where the "single thing" is a pair of objects, and the "rule" checks whether the target of the first equals the source of the last.)
