If we're trying to understand the holes in a space $X$, I guess one way to proceed would be to associate to $X$ a lax functor $\tilde{X} : \mathbb{N} \rightarrow \mathbf{Rel}$ defined as follows. Firstly, if $n \in \mathbb{N}$, then $\tilde{X}(n)$ is $\mathbf{Top}(S^n,X)$ modulo homotopy equivalence. Secondly, if $p:a \leq b$, then $\tilde{X}(p)$ is the relation $\tilde{X}(a) \nrightarrow \tilde{X}(b)$ defined as follows: $(f,g) \in \tilde{X}(p)$ iff there exists a realization $F : S^a \rightarrow X$ of $f$ and a realization $G : S^b \rightarrow X$ of $g$ such that there exists an embedding $\varphi : S^a \rightarrow S^b$ such that $G \circ \varphi = F$.
Question. This seems like a pretty reasonable variant on the more usual "simplicial set of simplices in $X$" (or whatever it's called.) Is this line of thinking pursued anywhere? If not, are there any technical problems with my definition that would limit its naturalness or usefulness?
