I know that the higher homotopy groups $\pi_n(X)$ can be defined in two way.
Homotopy classes of the continuos maps $(I^n,\partial I^n)\longrightarrow (X,x_0)$.
Homotopy classes of the continuos maps $(S^n,s_0)\longrightarrow (X,x_0)$.
These two ways to define the higher homotopy groups are equivalent. We can see this by using the universal proprerty of the quotient topological space. My question is...Is there a categorical meaning of the equivalence of these two definitions?
There is a background to this use of either cubes or spheres, but it needs a more general setting which is given in this paper Modelling and Computing Homotopy Types: $1$ to appear in Indagationes Math. in 2017.
One distinction between the use of cubes and spheres is that the composition or gluing of cubes is very clear, while the composition for homotopy groups defined by spheres is less direct. In a general setting, this can be seen as broad and narrow methods, and both are needed in the subject.
This distinction is not easy to see in the use of groups, although the use of various geometric models (cells, simplices, cubes, ...) has been standard in homotopy theory almost since its inception. The distinction becomes necessary when one moves to a many base point viewpoint, and tries to involve forms of higher groupoids.
The cited paper contains my views and may not be shared generally!