Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.
Question: How do you prove that the following diagram of homotopy groups commutes?:
$\pi_n(Y) \to \pi_{n-1}(\Omega Y)$
$\ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow$
$\pi_{n-1}(F) \to \pi_{n-2}(\Omega F)$
Admittedly, I don't know for certain that it commutes, but it looks like it should.
All the arrows are boundary maps ($\delta$) from long exact sequences of homotopy groups for a fibration. The horizontal maps are isomorphisms.
The definition of $\delta$ that I'm using is that for for a fibration $X \to Y$, $\alpha \in \pi_n(Y)$, $\delta ([\alpha]) = [\beta d^0]$, where $\beta:\Delta^n \to X$ fits into the following diagram:
$ \Lambda^n_0 \to^* X$
$\ \downarrow \ \ \ \ \ \downarrow$
$\Delta^n \to^\alpha Y$
So far, I've attempted to chase elements around this diagram and use prismatic arguments, but I haven't found one that works.
$\Omega F \to \Omega X \to \Omega Y$
$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$
$PF \to PX \to PY$
$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$
$\ \ F \to \ \ X \to \ \ Y$
It feels like a useful fact that $PX \to PY\times_Y X$ is a fibration, which is true since pointed simplicial sets form a simplicial model category.
Edit: The path and loop spaces I'm using are defined by the following: For pointed simplicial sets X and Y, define the simplicial set $hom_*(X,Y)$ to have n- simplices $hom_{sSet_*}(X \wedge \Delta^n_+,Y)$, where $\Delta^n_+$ is the standard n-simplex with a disjoint basepoint.
Then I'm using $PX=hom_*(\Delta^1 , X)$
and $\Omega X=hom_*(\Delta^1/\partial \Delta^1,X)$
