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This is joint work with Robert Boltje and Merve Demirel. A linear category is a category such that the sets of morphisms are modules and the composition is bilinear. We shall be concerned with cases where the isomorphism classes are partially ordered and all morphisms are linear combinations of morphisms which factor downwards and then upwards. In these cases, we shall classify the simple functors from the given category to the category of modules of the coefficient ring. We shall specialize this to scenarios where the objects are finite groups and the morphisms are associated with linear maps such as inductions, restrictions, inflations.