Categories of Bott-Samelson varieties

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We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s) BS(s'), where s is a subsequence of s'. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded Hₜ(pt)-modules coinciding on the objects with the usual functor Hₜ of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).