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My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature. Let me be more precise about what I mean. ...

For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper, https://www.uni-due.de/~bm0032/publ/TateMotives.pdf There exists an abelian category of mixed Tate ...

The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf states that the functor induced by $hs:\text{ECM} \rightarrow \text{...