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As the previous answer points out you have to consider local systems for a finer topology than the Zariski topology. It is natural to consider the étale topology. The category of étale local systems o …

You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the str …

Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set up of a Riemann surface with cusps. They were trying to generalize the Narasimhan-Seshadri correspondence on a compact Ri …

If a (say constant) group $G$ acts on a scheme $X$, you may want to consider the notion of a $G$-sheaf : a sheaf $\mathcal F$ endowed with isomorphisms $\lambda_g: g^* \mathcal F\simeq \mathcal F$, fo …

There is this very beautiful survey Nakamura, Hiroaki; Tamagawa, Akio; Mochizuki, Shinichi The Grothendieck conjecture on the fundamental groups of algebraic curves http://www.math.sci.osaka-u.ac. …

It depends on what you call Nori's fundamental group scheme, of course. Nori himself has given several versions of his fundamental group scheme, and it has been vastly generalized. If you think of th …

The source text Kato, Kazuya Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989 …

The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online: Tag04Y1 in the Stacks Project. I quote: Lemma 8.4. Let $C$ be a site. Let $f : X \to Y$ …

This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is q …

About your main question I suggest looking at Appendix A in On Nori's Fundamental Group Scheme Hélène Esnault, Phùng Hô Hai, Xiaotao Sun Geometry and dynamics of groups and spaces, 377–398, Progr. M …

As Keerthi Madapusi Pera points out in his comments, it is certainly reasonable to define a unipotent flat vector bundle as a flat vector bundle that is a successive extension of the trivial one $(\m …

The answer is "yes" I think, even if you replace $\mathbb Q$ with a number field. In the affine case this is a result of Matsumoto, as pointed out by Felipe Voloch, see Matsumoto, Makoto Galois rep …

The obstruction to the existence of such an isomorphism is a (bi)torsor, that has been studied in various real-life situations. An example extracted from On the relation between Nori Motives and Kon …

Besides the references already given above, one can also mention §7 of SGA III, Exp.X (Caracterisation Et Classification Des Groupes De Type Multiplicatif, doi:10.1007/BFb0059008), where Grothendieck …

The answer is yes (with mild hypothesis on the space). Moreover the topological situation is simpler, and this was very likely Grothendieck's inspiration. To see this you need two facts. First take …