Categories of Bott-Samelson varieties

We consider all Bott-Samelson varieties ${\rm BS}(s)$ 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 ${\rm BS}(s)\hookrightarrow{\rm BS}(s')$, 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 $H^\bullet_T({\rm pt})$-modules coinciding on the objects with the usual functor $H_T^\bullet$ 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).