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Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty}$$\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in 'Galois"Galois Groups over $\mathbb{Q}$'"), defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat connection and the crystalline equivalent given by unipotent overconvergent isocrystals, whereas the automorphism groups of their fibre functors define the $\ell$-adic, de Rham and crystalline fundamental groups respectively. Now, I do not claim to have a deep understanding of these constructions, nor of the paper as a whole (or to even have read it fully in detail, given its size!), but I was left with a question: why does the word unipotent appear in all the constructions?

Having read parts of Minhyong Kim's work it is understandable that in this context, these subcategories generated by the unipotent objects are not only enough, but seemingly exactly what is needed. In other words, one still acquires the Tannakian structure (which if I'm not mistaken exists on the full categories as well) and thus has fundamental groups which are affine group schemes and can start playing the game of using unipotency and quotients in the lower central series to construct moduli spaces of path torsors, moving gradually away from abelianness and deeper into Diophantine information. This however doesn't answer the original motivation.

I apologise if the answer is hidden in Deligne's original paper or if the context makes it obvious to everyone but me.

Deligne in his 1987 paper on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty}$ (in 'Galois Groups over $\mathbb{Q}$') defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat connection and the crystalline equivalent given by unipotent overconvergent isocrystals, whereas the automorphism groups of their fibre functors define the $\ell$-adic, de Rham and crystalline fundamental groups respectively. Now, I do not claim to have a deep understanding of these constructions, nor of the paper as a whole (or to even have read it fully in detail, given its size!), but I was left with a question: why does the word unipotent appear in all the constructions?

Having read parts of Minhyong Kim's work it is understandable that in this context, these subcategories generated by the unipotent objects are not only enough, but seemingly exactly what is needed. In other words, one still acquires the Tannakian structure (which if I'm not mistaken exists on the full categories as well) and thus has fundamental groups which are affine group schemes and can start playing the game of using unipotency and quotients in the lower central series to construct moduli spaces of path torsors, moving gradually away from abelianness and deeper into Diophantine information. This however doesn't answer the original motivation.

I apologise if the answer is hidden in Deligne's original paper or if the context makes it obvious to everyone but me.

Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat connection and the crystalline equivalent given by unipotent overconvergent isocrystals, whereas the automorphism groups of their fibre functors define the $\ell$-adic, de Rham and crystalline fundamental groups respectively. Now, I do not claim to have a deep understanding of these constructions, nor of the paper as a whole (or to even have read it fully in detail, given its size!), but I was left with a question: why does the word unipotent appear in all the constructions?

Having read parts of Minhyong Kim's work it is understandable that in this context, these subcategories generated by the unipotent objects are not only enough, but seemingly exactly what is needed. In other words, one still acquires the Tannakian structure (which if I'm not mistaken exists on the full categories as well) and thus has fundamental groups which are affine group schemes and can start playing the game of using unipotency and quotients in the lower central series to construct moduli spaces of path torsors, moving gradually away from abelianness and deeper into Diophantine information. This however doesn't answer the original motivation.

I apologise if the answer is hidden in Deligne's original paper or if the context makes it obvious to everyone but me.

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Unipotency in realisations of the motivic fundamental group

Deligne in his 1987 paper on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty}$ (in 'Galois Groups over $\mathbb{Q}$') defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat connection and the crystalline equivalent given by unipotent overconvergent isocrystals, whereas the automorphism groups of their fibre functors define the $\ell$-adic, de Rham and crystalline fundamental groups respectively. Now, I do not claim to have a deep understanding of these constructions, nor of the paper as a whole (or to even have read it fully in detail, given its size!), but I was left with a question: why does the word unipotent appear in all the constructions?

Having read parts of Minhyong Kim's work it is understandable that in this context, these subcategories generated by the unipotent objects are not only enough, but seemingly exactly what is needed. In other words, one still acquires the Tannakian structure (which if I'm not mistaken exists on the full categories as well) and thus has fundamental groups which are affine group schemes and can start playing the game of using unipotency and quotients in the lower central series to construct moduli spaces of path torsors, moving gradually away from abelianness and deeper into Diophantine information. This however doesn't answer the original motivation.

I apologise if the answer is hidden in Deligne's original paper or if the context makes it obvious to everyone but me.