Homotopy-group local system along a cellular map

Let $n\ge 2$, let $X$ be a CW complex, and let $f:X^n\to Y$ be cellular. On $X^n$ define the homotopy-group local system along $f$ by

$(f^{\ast }{\mathrm{\Pi }}_{n}Y{)}_x={\pi }_n(Y,f(x)),T_{\gamma }={\beta }_{\overline{f\gamma }}:{\pi }_n(Y,f(x))⟶{\pi }_n(Y,f(y))$

for a path class $\gamma :x\to y$. The reversal is forced by the published basepoint-transport convention, in which ${\beta }_{\rho }:{\pi }_n(Y,\rho (1))\to {\pi }_n(Y,\rho (0))$. Endpoint-fixed homotopy invariance and ${\beta }_{\rho \ast \lambda }={\beta }_{\rho }{\beta }_{\lambda }$ give $T_{\gamma \ast \eta }=T_{\eta }{T}_{\gamma }$, so this is a covariant functor to abelian groups.

The pair $(X,X^n)$ has only cells of dimension at least $n+1\ge 3$. The published high-relative-cell lemma therefore shows that ${\mathrm{\Pi }}_1(X^n)\to {\mathrm{\Pi }}_1(X)$ is an equivalence: it is bijective on components and induces isomorphisms on all vertex groups. Consequently $f^{\ast }{\mathrm{\Pi }}_{n}Y$ extends to a local system on $X$, uniquely up to a natural isomorphism whose restriction to $X^n$ is the identity. An obstruction calculation must either fix one such extension as coefficient data or use the equivalent universal-cover module model. For a point outside $X^n$ its stalk is not written ${\pi }_n(Y,f(x))$, since $f(x)$ is not defined there. This corrects the ill-typed wording in the Step-1 scaffold.

For $n=1$, this page uses the construction only when the relevant ${\pi }_1(Y)$ is abelian and all conjugation transport is trivial. Then the system has trivial monodromy and is isomorphic to a constant abelian system on each component. No nonabelian group is inserted into a cellular cochain group. The definition itself chooses neither component basepoints nor a set-indexed family of paths; any concrete coordinate extension is treated as supplied data.