I found in some articles a definition of the Galois group of a differential module. In the definition appeared the expression $(Repr(Gal(M,\nabla),Forget)$. According to the the articles, this pair should be equivalent to $(C(M,\nabla),F)$ where $(M,\nabla)$ - fiber bundle with a connection, $C(M,\nabla)$ - tensor constructions of $(M,\nabla)$ ,$Rep(Gal(M,\nabla))\subset Gl(n,\mathbb{C})$ - representation of the Galois group, $Sol(\nabla)$ - space of horizontal sections of $\nabla$ and $F:(M,\nabla) \to Sol(\nabla) $ - a functor. My question is: which is $Forget$? I could't find any descritpion of this term.
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$\begingroup$ "Forget" presumably denotes a forgetful ("fiber") functor, the thing you take automorphisms of in the context of Tannakian reconstruction. $\endgroup$– Qiaochu YuanOct 2, 2015 at 15:04
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As Qiaochu Yan points out, it means forgetful.
When applied to a differential field $K$, the neutral Tannakian category $\mathrm{Diff}_K$ of differential modules over $K$ comes with a forgetful fibre functor $\rho :\mathrm{Diff}_K \to \mathrm{Vect}_K$ that associates the $K$-vector space $M$ to the differential module $(M, \partial)$.
In this context, the subcategory $\{\{M\}\}$ of $\mathrm{Diff}_K$ is equivalent to $\mathrm{Repr}_G$, $G$ the differential Galois group of $M$ over $K$.
