Local systems and pullback

Fix a commutative unital ring $R$. A left $R$-module local system on a space $X$ is a covariant functor $\mathcal{L}:{\mathrm{\Pi }}_1(X)⟶R\text{-}\mathbf{M}\mathbf{o}\mathbf{d},$ where the source is Fundamental groupoid of a space, functor means Covariant functor, identity functor, composite functor, and contravariant functor, and the target is the category in Left modules over a fixed ring and module homomorphisms form the large locally small category $R\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$. Write ${\mathcal{L}}_x$ for its value at $x$ and $T_{\gamma }=\mathcal{L}([\gamma ]):{\mathcal{L}}_x⟶{\mathcal{L}}_y$ for transport along a path $\gamma :x\to y$. Every arrow of the source has a reversed inverse, so functoriality makes every $T_{\gamma }$ an isomorphism, with $T_{\bar{\gamma }}=T_{\gamma }^{-1}$ and $T_{\alpha \ast \beta }=T_{\beta }{T}_{\alpha }$.

A morphism of local systems $\eta :\mathcal{L}\to \mathcal{K}$ is a natural transformation (Natural transformation and its components): it is a family of $R$-linear maps ${\eta }_x:{\mathcal{L}}_x\to {\mathcal{K}}_x$ satisfying ${\eta }_y{T}_{\gamma }^{\mathcal{L}}=T_{\gamma }^{\mathcal{K}}{\eta }_x$ for every path $\gamma :x\to y$. Isomorphisms of local systems are natural isomorphisms.

A continuous map $f:X\to Y$ induces a functor ${\mathrm{\Pi }}_1(f):{\mathrm{\Pi }}_1(X)\to {\mathrm{\Pi }}_1(Y)$ by $x\mapsto f(x)$ and $[\gamma ]\mapsto [f\gamma ]$. Postcomposition preserves constants, reversals, and concatenation, so this is well defined. The pullback local system is $f^{\ast }\mathcal{K}=\mathcal{K}\circ {\mathrm{\Pi }}_1(f),(f^{\ast }\mathcal{K}{)}_x={\mathcal{K}}_{f(x)},T_{\gamma }^{f^{\ast }\mathcal{K}}=T_{f\gamma }^{\mathcal{K}}\mathrm{.}$ Pullback of a coefficient morphism is defined componentwise. Thus $(gf{)}^{\ast }=f^{\ast }{g}^{\ast }$ and $1_X^{\ast }$ are literal equalities of functors with these conventions.

All definitions work independently on every path component. The empty space has the unique empty local system. No basepoint, universal cover, common fiber, or choice principle is required.

For later comparison with group rings, if $x$ is fixed and $g=[\alpha ]\in {\pi }_1(X,x)$, the base fiber is given the left monodromy convention $g\cdot m:=T_{\bar{\alpha }}(m)\mathrm{.}$ The reversal is essential: covariance and first-path-first multiplication give $T_{\alpha \ast \beta }=T_{\beta }{T}_{\alpha }$, so $T_{\overline{\alpha \ast \beta }}=T_{\bar{\alpha }}{T}_{\bar{\beta }}$, exactly the left action law $(gh)m=g(hm)$.