# Based loops objects in model categories

If I have a pointed model category then for I can define based loops objects as homotopy pullbacks:

$$\require{AMScd}$$ $$\begin{CD} \Omega X @>>> *\\ @V V V @VV V\\ * @>>> X \end{CD}$$

If my category is additionally powered by simplicial sets, how can I realize iterated loop objects through maps from spheres? I guess this has to be straightforward but I cannot see why it works.

• This ultimately boils down to your preferred definition of spheres. Let S^0 denote ∆^0 with a disjoint basepoint. If one defines S^1 to be the homotopy pushout in simplicial sets of the diagram * <- S^0 -> *, then you basically need to show that S^1 smashed with itself n times is the same as your favorite model for an n-sphere. – skd Sep 19 '19 at 22:23