  In this paper we study categories of tilting modules. Our starting point is
the tilting modules for a reductive algebraic group G in positive
characteristic. Here we extend the main result in [8] by proving that these
tilting modules form a (strictly object-adapted) cellular category. We use this
result to specify a subset of cellular basis elements, which generates all
morphisms in this category. In a different direction we generalize the earlier
results to the case where G is replaced by the infinitesimal thickenings G_rT
of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here our
procedure leads to a special set of generators for the morphisms in the
category of projective G_rT- modules. Our methods are rather general (applying
to "quasi hereditary like" categories). In particular, there are completely
analogous results for tilting modules of quantum groups at roots of unity. As
examples we treat the tilting modules in the ordinary BGG category O, and in
the modular case we examine G = SL_2 in some details.
