Let $X$ be a scheme which is smooth over a noetherian base scheme $S$.
Let $(\mathcal F,\nabla)$ be a flat vector bundle, i.e. $\mathcal F$ is a vector bundle of finite rank on $X$ and $\nabla$ is an integrable connection relative $S$.


Is there a notion in the literature of "unipotent" flat vector bundle and morphisms between them? 
And does this give an abelian category with objects isomorphism classes of such $(\mathcal F, \nabla)$? I haven't seen a clean formal approach to this in the literature.


I don't think it is abelian in the above wide context, but probably for $X$ being a smooth variety over a field or an abelian variety over a field.



Unipotent vector bundles
As [Keerthi Madapusi Pera](https://mathoverflow.net/users/7868/keerthi-madapusi) 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 $(\mathcal O\_X,d)$. Over a general basis $S$ I don't know, but over a field there are plenty of references.


For instance let $k$ be a field of characteristic zero and $X$ a smooth geometrically connected scheme over $k$. Then according to


Deligne, P.  

[Le groupe fondamental de la droite projective moins trois points.](https://www.math.ias.edu/files/deligne/GaloisGroups.pdf)  

[Zbl 0742.14022](https://zbmath.org/0742.14022)


$\S$ 10.26 the category of unipotent flat vector bundles over $X$ is even Tannakian (a fortiori abelian), giving rise to the "De Rham fundamental group" $\pi(X,x)\_{DR}$.


If you want to work in positive characteristic, or with singularities, flat vector bundles are certainly not the right objects any longer (at least if you want to have some link with the fundamental group), one has to replace them with stratifications, see


Saavedra Rivano, Neantro  

[Catégories Tannakiennes.](https://doi.org/10.1007/BFb0059108)  

Lecture Notes in Mathematics, Vol. 265.  

[Zbl 0241.14008](https://zbmath.org/0241.14008)


VI 1.2


or


dos Santos, João Pedro Pinto  

[Fundamental group schemes for stratified sheaves](https://doi.org/10.1016/j.jalgebra.2007.03.005)  

J. Algebra 317 (2007), no. 2, 691–713.  

[Zbl 1130.14032](https://zbmath.org/1130.14032)


One could also mention that over a complete scheme over a field, you don't need to consider connexions at all. For instance Nori in chapter IV of his PhD


Nori, Madhav V.  

[The fundamental group-scheme.](https://doi.org/10.1007/BF02967978)  

Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122.  

[Zbl 0586.14006](https://zbmath.org/0586.14006)


considers a scheme of finite type $X$ over a field $k$ such that $H^0(X,\mathcal O\_X)=k$, and proceeds to show that the category of unipotent vector bundles is indeed Tannakian (the explanation is that one can deduce from this that unipotent vector bundles are in fact endowed with connections). Lemma 2 there may be of interest for you, because he shows exactly the abelianness in this similar context.


