For simplicity here only the affine (affinoid) case is written here but the framework work with general case.
Unipotent Isocrystals
A unipotent isocrystal on \(\overline{A}\) is an \(A^\dagger_K\)-module \(M\) and an integrable connection
\[ \nabla : M \to M\otimes_{A^\dagger_K} (\Omega^1_{A^\dagger} \otimes K) \]
obtained by iterated extensions of trivial connections \((A^\dagger_K, d)\). In this case the extension is actually a free module.
A morphism of unipotent isocrystals is a morphism of \(A^\dagger\)-modules \(f:M\to N\) such that \(\nabla_N \circ f = (f\otimes \Omega^1_K) \circ \nabla_M\). Here \(\Omega^1_K = \Omega^1_{A^\dagger} \otimes K\).
The category of unipotent isocrystals on \(\overline{A}\) is denoted by \(\mathcal{U}(\overline{A})\), which as a category only depends on \(\overline{A}\) independent of the choice of lift.
Example of Notations
\(\mathcal{T}_n\) is the Tate algebra
\[\mathcal{T}_n := \left\{ \sum a_I t^I : a_I\in R, |a_I|\to 0 \right\}\]
\(A^\dagger\) is the weakly complete finitely generated (wcfg) algebra, a surjective image of Tate algebra
\[A^\dagger := \mathcal{T}_n / (f_1,\dots,f_m)\]
\(\mathcal{T}_n^\dagger/\pi = \kappa[t_1,\dots,t_m]\) is the polynomial ring obtained under reduction, so is \(\overline{A} = A^\dagger/\pi = \kappa[t_1,\dots,t_m]/(\overline{f_1},\dots,\overline{f_m})\) a finitely generated \(\kappa\)-algebra since it is a quotient of Tate algebra.
The overconvergent differential module \(\Omega^1_{A^\dagger}\) is not the algebraic module of differential, rather it is the overconvergent module of differentials, which is a \(A^\dagger\)-module given by
\[\Omega^1_{A^\dagger} := \bigoplus_{i=1}^n A^\dagger \,\mathrm{d}{t_i} / (df_1,\dots,df_m).\]
The point is that we need to take limit, and in algebraic differential, \(\mathrm{d}\) does not cooperate with limit.
Example
Any rank two unipotent isocrystal \(M\) is an extension in \(\mathrm{Ext}^1_{\mathcal{U}}(1,1)\). This splits due to freeness (non-canonical), so it is isomorphic to some \((\mathcal{O}^2, \nabla)\). Let’s see what happens inside, writing out
If we denote the injection \(\mathcal{O}_X\to M\) giving out the first frame section \(e_1\), and choose a non-canonical section that inverts the surjection \(M\to \mathcal{O}_X\) which maps \(e_2\), then the vertical arrows says
\(\nabla (e_1) = 0\) because the injection commutative diagram
\(\nabla(e_2) = 0\) when you mod out \(e_1\), so \(\nabla(e_2) = \omega e_1\) for some \(\omega\).
The requirement that \(\nabla\) is a flat connection gives \(\nabla(\omega e_1) = d\omega \wedge e_1=0\), so \(\omega\) is a closed form.
We have found out that \(\nabla = d\cdot I_2 + \Lambda\) where
\[ \Lambda = \begin{pmatrix} 0 & \omega \\ 0 & 0 \end{pmatrix} \]
In fact this correspondence gives a bijection
\[ \mathrm{Ext}^1_{\mathcal{U}(\overline{A})}(1, 1) \cong H^1_{MW}(\overline{A}/K),\quad [\Lambda] \mapsto [\omega]. \]
In Detail
Let’s consider why the above morphism \(\varphi\) is an isomorphism in detail. This means
It is well-defined, i.e. isomorphic unipotent isocrystals give the same cohomology class.
It is surjective, luckily this is obvious because \(\omega\) can be any closed form.
It is injective, i.e. if \(\omega=\,\mathrm{d}f\) is exact, then the corresponding \(\Lambda\) gives a connection that splits, it is isomorphic to a connection whose matrix is semi-simple.
When split happens, we have a map \(\mathcal{O}\to M: e_2\mapsto ae_1+e_2\), where
\[\begin{align*} ae_1+e_2 & \leftarrow e_2 \\ (\mathrm{d}a + \omega) e_1 & \leftarrow 0 \\ \end{align*}\]
i.e. we will have \(\omega = - \mathrm{d} a\). Conversely if \(\omega\) is exact we can construct the split. Thus \(\omega\) is exact iff the extension splits.
Tensor Structure
The category \(\mathcal{U}(\overline{A})\) is a rigid abelian tensor category, and it can be made into a Tannakian category with the fibre functors. Let \(x\in X_\kappa(\kappa)\) be a rational point in reduction, the fibre functor at \(x\) is a functor from \(\mathcal{U}(\overline{A})\) into \(K\)-vector spaces of flat sections over a local residue disk \(U_x\) of \(x\),
\[ \omega_x : \mathcal{U}(\overline{A})\to \mathrm{Vec}_K, \quad \omega_x[(M, \nabla)] := \{ v\in M(U_x), \nabla v = 0 \}. \]
Overconvergence is required to make \(\dim M(U_x) = \mathrm{rank}M\). Finding horizontal sections for a unipotent connection is the same as iterated integration, which is feasible because of overconvergence.
\(\omega_x\) can also be viewed as a pullback \(x^*\) to an isocrystal on \(\mathrm{Spec}(\kappa)\).
The category \(\mathcal{U}(\overline{A})\) together with fibre functor \(\omega_x\) determines an affine pro-algebraic fundamental group
\[ G = G_x = \pi_1(\mathcal{U}(\overline{A}), \omega_x) \]
whose functor of points over any \(K\)-algebra \(F\), \(G(F)\) is given by
\[G(F) = \mathrm{Aut}_{\otimes}(\omega_x \otimes F),\]
here \(\omega_x \otimes F\) is the functor \((\otimes F)\circ \omega_x\), The group \(\mathrm{Aut}_{\otimes}(\omega_x\otimes F)\) is the group of natural automorphisms for the functor \(\alpha: \omega_x \otimes F \to \omega_x \otimes F\) that commutes with the tensor structure, i.e. \(\alpha_{M\otimes N} = \alpha_M \otimes \alpha_N\) and \(\alpha_{\mathcal{O}} = \mathrm{id}\).
Written explicitly,
\[\begin{align*} G_x(F) &= \mathrm{Aut}_\otimes(\omega_x \otimes F)\\ &= \,\,\,\{\,\forall M\in \mathcal{U}(\overline{A})\\ & \quad\quad, \,\alpha_M: \omega_x(M)\otimes F \simeq \omega_x(M)\otimes F \in \mathrm{NatIsom}({}_{F}\mathbf{Mod})\\ & \quad\quad, \,\alpha_{M\otimes N} = \alpha_M \otimes \alpha_N\\ & \quad\quad, \, \alpha_{1} = \mathrm{id}\\ &\quad\quad \} \end{align*}\]
There is an equivalence of categories (similar to Riemann-Hilbert correspondence) of finite dimensional \(K\)-representations of \(G\) and unipotent isocrystals on \(\overline{A}\)
\[ \mathrm{Rep}_K(\pi_1(\mathcal{U}(\overline{A}),\omega_x)) \simeq \mathcal{U}(\overline{A}) \]
Lie Algebra Of The Fundamental Group
To extract the Lie algebra of \(G(\cdot)\), which is the tangent plane \(T_eG\) of \(G\) at identity, we can use the fact that
\[T_xX = \mathrm{Hom}_{\mathrm{Spec}(K[\varepsilon]/\varepsilon)\mapsto x}(\mathrm{Spec}(K[\varepsilon]/\varepsilon^2), X).\]
Let subscript \(\otimes\) denote all the tensor laws \(\alpha_{M\otimes N} = \alpha_M\otimes \alpha_N\) and \(\alpha_{1} = \mathrm{id}\), we have
\[\begin{align*} \mathfrak{g}&= T_eG\\ &= \mathrm{Hom}_{\mathrm{Spec}(K[\varepsilon]/\varepsilon)\mapsto e}(\mathrm{Spec}(K[\varepsilon]/\varepsilon^2), G) \\ &= \left[G(K[\varepsilon]/\varepsilon^2) \xrightarrow{\mathrm{ev}_{\varepsilon=0}} G(K)\right]^{-1}(e) \\ &= \ker\left[G(K[\varepsilon]/\varepsilon^2) \xrightarrow{\mathrm{ev}_{\varepsilon=0}} G(K)\right] \\ &= \ker\left[\{ \alpha_M : \omega_x(M)\otimes K[\varepsilon] \to \omega_x(M)\otimes K[\varepsilon]\}_\otimes \xrightarrow{\mathrm{ev}_{\varepsilon = 0}} G(K)\right] \\ &= \left\{ \alpha_M = \begin{pmatrix} 1_{\omega_x(M)} & 0 \\ \varepsilon\beta_M & \varepsilon 1_{\omega_x(M)} \\ \end{pmatrix} \right\}_\otimes \\ &= \left\{ \alpha_M = 1_{\omega_x(M)\otimes K[\varepsilon]} + \varepsilon\beta_M: \beta_M \in \mathrm{End}(\omega_x(M)) \right\}_\otimes \\ &= \left\{ \beta_M \in \mathrm{End}(\omega_x(M)) \right\}_\otimes \end{align*}\]
By standard arguments similar to differentiation, we see that tensor laws \(\alpha_{M\otimes N} = \alpha_M \otimes \alpha_N\) and \(\alpha_{1} = \mathrm{id}\) are equivalent to
\(\beta_{M\otimes N} = \beta_M \otimes 1_{\omega_x(N)} + 1_{\omega_x(M)}\otimes \beta_N\) and
\(\beta_1 = 0\).
Since \(G\) is unipotent, the Lie algebra \(\mathfrak{g}\) is nilpotent.
The exponential map \(\mathfrak{g}\to G\) is therefore algebraically defined.
This gives us an equivalence of categories of algebraic representations of \(G\) and continuous representations of \(\mathfrak{g}\).
\[\mathrm{Rep}(G) \simeq \mathrm{Rep}(\mathfrak{g})\]
The Frobenius Invariant Path
Given two \(\kappa\)-rational points \(x,z\in X_\kappa\), the space of functor isomorphisms is what we abstractly call a path space
\[ P_{x,z} := \mathrm{Isom}_\otimes(\omega_x, \omega_z) \]
which is a homogeneous space for the fundamental group \(G\), has a natural action of \(G_x^{op}\times G_z\), or left action by \(G_z\) and right action by \(G_x\). These ’path’s gives analytic continuations identifying horizontal sections in different local residue disks \(M(U_x)^\nabla\) and \(M(U_z)^\nabla\).
The Frobenius invariant path space is the subspace of \(P_{x,z}\) that is fixed by the action of the Frobenius automorphism \(\varphi\) of \(\overline{A}\).
Reference
Besser, Heidelberg Lectures on Coleman Integration