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The concept of a local system in algebraic geometry is often described as a locally constant constructible sheaf on a scheme $X$, which is in essence a sheaf whose stalk at a point $x$ comes equipped with an action of $\pi_1(X,x)$, where the fundamental group's definition is implicit in the category of local systems we are working on.

I guess that it is possible to consider, similarly, a sheaf whose stalks come equipped with an action of an abstract group, or perhaps a higher homotopy group $\pi_k(X,x)$. For some $k$. For whatever reason, I never encountered the latter in my studies. Are they important? Are they used somewhere in the literature?

Sorry for the vague question.

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Infinity local systems have been defined and studied by various people. For instance, Lurie gives the following definition in some notes Algebraic K-theory and Manifold Topology, notes for course Math 281

Let $X$ be a topological space and $\mathcal{C}$ be an $\infty$-category. A local system on $X$ with values in $\mathcal{C}$ is a map of simplicial sets, $$Sing(X)\to \mathcal{C}.$$ The collection of all local systems on $X$ with values in $\mathcal{C}$ then gives an $\infty$-category, $Fun(Sing(X),\mathcal{C})$.

Of course, this requires some background reading on $\infty$-categories. It is also easy to generalise this by replacing the $\infty$-groupoid, $Sing(X)$ by that of smooth singular simplices or a triangulation of $X$. There is some interesting material also in The higher Riemann-Hilbert correspondence and principal 2-bundles by Camilo Arias Abad and Sebastian Velez Vasquez, and A Riemann–Hilbert correspondence for infinity local systems by Jonathan Block and Aaron M. Smith (which is on the ArXiv).

There are ways of extending these topological constructions to an algebraic geometry setting, but I do not have the precise references with me.

Elsewhere a link with constructible sheaves, etc, has been explored following ideas on stratified spaces following ideas of Treumann, Exit paths and constructible stacks. Compos. Math. 145 (2009), no. 6, 1504–1532. That also leads to other very interesting constructions and generalisations, but I will not try to explain them here, unless that is of interest to someone as it does seem a bit further from the Alg. Geom context.

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    $\begingroup$ Local systems on $X$ can also be identified with locally constant sheaves of spectra on $X$; naive $\Omega X$ spectra; and $\Sigma^\infty_+ \Omega X$ modules. I think each point of view has its advantages, so best to consider them all together. $\endgroup$
    – Connor Malin
    Commented Jan 11, 2023 at 17:49
  • $\begingroup$ @ConnorMalin Is that the same as the definition in this answer? I suspect that you need something like $X$ has singular shape? $\endgroup$
    – Z. M
    Commented Jan 11, 2023 at 18:13
  • $\begingroup$ @Z.M I'm assuming $X$ is a CW complex here, thanks for pointing that out. $\endgroup$
    – Connor Malin
    Commented Jan 11, 2023 at 18:46
  • $\begingroup$ @Z.M See in Luries L-Theory Notes (math.ias.edu/~lurie/287xnotes/Lecture21.pdf) or this paper on Morita Cohomology (arxiv.org/abs/1404.1327) for a reference. $\endgroup$
    – Markus Zetto
    Commented Jan 11, 2023 at 21:20
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    $\begingroup$ @MarkusZetto As Connor Malin said above, it only applies to good spaces (although now I am not sure whether being of singular shape is enough, it is enough to assume to be locally of singular shape instead, cf. Higher Algebra, the paragraph before Cons A.4.17). $\endgroup$
    – Z. M
    Commented Jan 11, 2023 at 21:33

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