We explain various convex cones of positive maps between matrix algebras including entanglement breaking, k-superpositive, completely positive, k-positive maps, and corresponding convex cones of bi-partite states through Choi matrices and duality.
We also exhibit an identity which connect composition and tensor product of linear maps, with which we recover various criteria in a unified framework. We also discuss several open questions on the inclusions between decomposability and k-positivity.