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
$\Omega X=hom\_\*(\Delta^1/\partial \Delta^1,X)$



Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces
The boundary map of homotopy groups is induced by a map $\partial : \Omega Y \to F$
of spaces; then the commutativity follows from the naturality of the
isomorphism $\pi\_n \circ \Omega \cong \pi\_{n+1}$.


