Weil cohomology theories can be considered as fibre functors from the category of motives. Given two such functors, we have an affine scheme of invertible natural transformations between them, and rational Hodge theory can be considered as providing $R$-valued points of this scheme (where $R$ is the coefficient ring). See this post, for example.

Can integral Hodge theory be viewed in a similar way? We are not over a field, so there might be Tannakian issues. Maybe it is pretty trivial, but then it would be nice if somebody provided full details.

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