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Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operator...
I think $X^{\dagger}$ must mean the dagger space associated to the weak formal scheme you get by taking the weak completion of the model $\mathcal{X}$ along the special fibre $\mathcal{X}_{k}$. Then $ …