I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.


I am aware that for a topological space $X$ and a point $x \in X$ the fundamental group $\pi\_1(X,x)$ acts on all homotopy groups $\pi\_n(X,x)$. In particular, this means the $\mathbf{Q}$-vector space $\mathbf{Q} \otimes\_\mathbf{Z}\pi\_n(X,x)$ is endowed with a linear action by $\pi\_1(X,x)$ and hence defines a local system of $\mathbf{Q}$-vector spaces on $X$. If $X$ is a (complex) manifold or something and one can appeal to tools akin to the Riemann-Hilbert correspondence and other interpretations of these objects, can one describe these explicitly? Are they 'tautological' local systems in some sense... ? Can anything be said about the *family* of local systems $(\mathbf{Q}\otimes\_\mathbf{Z}\pi\_n(X,x))\_{n \geq 2}$ and can these be recognised as something else?


I'm not sure my question makes much sense and I was just trying to satiate a curiosity :) Any words of wisdom are appreciated and thank you very much! :D


Note: I asked [this question on MSE](https://math.stackexchange.com/questions/4669233/local-systems-defined-by-higher-homotopy-groups) and thought it would be best move it here.



Local systems arising from higher rational homotopy groups
Let's say we have rigorously defined the notion of a family of objects in an $\infty$-category $C$ parametrized by a space $X$. There are several ways to do this: if we model $\infty$-categories by quasicategories, we can simply consider $\mathrm{Fun}(X,C)$; and in other models it might be easier to consider the category of Borel $\Omega X$ objects of $C$, which morally are the same by clutching constructions.


With these technicalities out of the way, to a space $X$ there is a parametrized pointed space $T$ which over a point $x \in X$ is the pointed space $ X \times \{x\}$ with basepoint $(x,x)$. In other words, it is given by $X \times X \xrightarrow{pr\_2} X$ with the section $X \xrightarrow{\Delta} X \times X$. Of course, we can fiberwise apply rationalization to get a parameterized rational pointed space $T\_\mathbb{Q}$ which over $x$ is $( X \times \{x\})\_\mathbb{Q}$. Now since there is an equivalence of (suitably connected) rational pointed spaces and (suitably connected) rational shifted differential graded Lie algebras with augmentations, I can pass between the parameterized objects of either of these categories. In the above situation, this yields a parametrized (shifted) differential graded Lie algebra with augmentation that we call $\pi\_\mathbb{Q} (T)$.


The fiber over $X$ is then the augmented dg Lie algebra associated to $(X \times \{x\})\_\mathbb{Q}$ whose homology (up to a shift) is $\pi\_\*(X \times \{x\}) \otimes \mathbb{Q}$. Hence, if we apply fiberwise the homology functor $\mathrm{Alg}^{\mathrm{aug}}\_{\mathrm{lie}[-1]}(\mathrm{dgVect}\_{\mathbb{Q}})\rightarrow \mathrm{Alg}^{\mathrm{aug}}\_{\mathrm{lie}[-1]}(\mathrm{grVect}\_{\mathbb{Q}})$ to our parameterized dg Lie algebra $\pi\_\mathbb{Q} (T)$, we arrive at your family of local systems.


If you are interested in potential applications of $\pi\_\mathbb{Q} (T)$ then if $X$ is a manifold one can attempt to do rational surgery theory with the knowledge of $\pi\_\mathbb{Q} (T)$, since the tangent microbundle is encoded in the diagonal $X \xrightarrow{\Delta} X \times X$. If $X$ is not a manifold, one could still look at the obstructions to it being rationally homotopy equivalent to a manifold since $\pi\_\mathbb{Q} (T)$ actually encodes the rationalization of the Spivak normal fibration.


