On Auslander's formula and cohereditary torsion pairs

For a small abelian category C, Auslander's formula allows us to express C as a quotient of the category mod - C of coherent functors on C. We consider an abelian category with the added structure of a cohereditary torsion pair tau = (T, F). We prove versions of Auslander's formula for the torsion-free class F of C, for the derived torsion-free, class D-b(F) of the triangulated category D-b(C) as well as the induced torsion-free class in the ind-category Ind - C of C. Further, for a given regular cardinal alpha, we also consider the category mod(alpha) - C of alpha-presentable objects in the functor category Fun(C-OP, Ab). Then, under certain conditions, we show that the torsion-free class can be recovered as a subquotient of mod(alpha) - C