Categorification of geometry

Question

I don't know if this idea is known, relevant or dumb, but I noticed that one could define abstract connectedness with groupoids. Let us forget about topology for a while, and let us think algebraically.

Let $G$ be a non empty groupoid. Call it connected if for any objects $x$ and $y$ in $Obj(G)$, $Arr(x,y)$ is non empty. A "connected" map is nothing else than a functor from connected groupoids. In particular, one could see any groupoid as a sum (coproduct) of connected groupoids.

Let $[0,r]$ be the real interval of length $r$ seen as a poset category. One can define a path of length $r$ as a functor $F: [0,r] \rightarrow G$. Clearly, given two functors $F_1: [0,r] \rightarrow G$, $F_2: [0,q] \rightarrow G$, if $F_2(0) = F_1(r)$, one can define their concatenation as $F_2 + F_1 : [0,r+q] \rightarrow G$ in the obvious way.

Say that two functors of the same length are homotopic if there exists a natural equivalence between them.

Moreover, it is clear that there exists constant functors of length $q$, such that one can define equivalence classes of paths (following the construction of Brown for homotopic maps) as follow: two functors $F_1$ and $F_2$ are equivalent if there exists constant functors $q$ and $r$ such that $r+F_1$ is homotopic to $q+F_2$.

This allows us to build the category $\Pi_1(G)$ which is a groupoid, whose objects are the objects of $G$ and the arrows are the equivalence classes of functors. Clearly, the compositions given as cls(F) + cls(G) = cls(F+G) is well defined, obviously associative, and admits as identity the class of the constant functors. Moreover, there exists an inverse for each class (again, given obviously by the class of the opposite functor of some representative).

One could even speak about higher "homotopy group" of groupoids (not necessarily connected) by defining paths from the circle (seen as a category, that is, the pushout in Cat $[0,1] \leftarrow \{0,1\} \rightarrow 1$) to $G$.

My question is, could we take groupoids seriously as the "base spaces", forgetting everything about topology? Why should we even care about topology, actually?

Answer 1

Yes, but you need to use some model of higher groupoids to capture the part of homotopy theory that can't be captured by groupoids. This is Grothendieck's homotopy hypothesis. Unfortunately, it's hard to write down models of higher groupoids, and in practice it's often (but not always!) the case that the easiest way to describe a higher groupoid to another human is to write down a topological space. That is, topological spaces are very convenient presentations of higher groupoids. But other convenient presentations are known, e.g. simplicial sets.

Answer 2

@Qiaochu: The situation is not as dire as you suggest: we can easily define the notion of $n$-fold groupoid, and it is true that these model all weak homotopy $n$-types. Some forms of these, for example when one of the groupoid structures is a group, have been well studied and applied. However they arise homotopically not from just plain spaces but from $(n-1)$-cubes of (pointed) spaces. For some background to these ideas, see this presentation (Galway, Dec, 2014), which explains how I was led away from spaces to certain structured spaces.

Not quite as complicated are some higher groupoid structures dealing with filtered spaces, as described in this 2011 EMS Tract Nonabelian Algebraic Topology.