Fiber homology local system of a Serre fibration

Let $p:E\to B$ be a Serre fibration, $R$ a commutative unital ring, and $q\ge 0$ an integer. Let $p_p:E_p\to B$ be the canonical mapping-path Hurewicz replacement, and let $j_b:F_b\to F_b^{\mathrm{\prime }}=(p_p{)}^{-1}(b)$ be the strict-fiber comparison. Write $J_{b,q}:H_q(F_b;R)\stackrel{\cong }{\to }{H}_q(F_b^{\mathrm{\prime }};R)$ for the isomorphism of Serre-fibration replacement preserves fiber homology transport. The fiber-homology local system is the functor ${\mathcal{H}}_q(p;R):{\mathrm{\Pi }}_1(B)⟶R\text{-}\mathsf{M}\mathsf{o}\mathsf{d}$ defined on objects by ${\mathcal{H}}_q(p;R)(b)=H_q(F_b;R)$ and on an endpoint-fixed path class $[\gamma :b\to c]$ by ${\mathcal{H}}_q(p;R)([\gamma ])=J_{c,q}^{-1}{H}_q(T_{\gamma };R)J_{b,q}\mathrm{.}$ This homological definition is choice-free. Its identity, path-homotopy, composition, and inverse laws are transported from the Hurewicz local system in Fiber transport gives the Serre local systems, with the underlying transport laws supplied directly by Fibers over one path component are fiber homotopy equivalent.

For cohomology, assume the Axiom of Choice as in The Axiom of Choice. Apply The universal coefficient theorem for cohomology over a PID over $\mathbb{Z}$ to the comparison $j_b$. Naturality, together with its integral-homology isomorphisms, makes $K_b=j_b^{\ast }:H^q(F_b^{\mathrm{\prime }};R)\stackrel{\cong }{\to }{H}^q(F_b;R)$ an isomorphism. The fiber-cohomology local system is ${\mathcal{H}}^q(p;R):{\mathrm{\Pi }}_1(B)⟶R\text{-}\mathsf{M}\mathsf{o}\mathsf{d},{\mathcal{H}}^q(p;R)(b)=H^q(F_b;R),$ with ${\mathcal{H}}^q(p;R)([\gamma :b\to c])=K_c{H}^q(T_{\bar{\gamma }};R)K_b^{-1}\mathrm{.}$ The reversed path compensates for cohomological contravariance, so the result is again covariant on the fundamental groupoid in the sense of Fundamental groupoid of a space and Local systems and pullback.

For a strictly commuting square of Serre fibrations, functoriality of the mapping-path construction and the Hurewicz systems gives a natural transformation in the forward direction for ${\mathcal{H}}_q$ and in the reverse coefficient direction for ${\mathcal{H}}^q$. Empty strict fibers give zero stalks and remain empty along their base component; point fibers and $q=0$ use the same formulas. If $R$ is the zero ring, both local systems are zero. No cohomological construction is claimed here without its stated AC hypothesis.