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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.

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    $\begingroup$ 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. $\endgroup$
    – skd
    Commented Sep 19, 2019 at 22:23

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