The paper describes the definition by generators and relations of an action of a braids monoid on a category. The main tool is a theorem on the contractibility of the set of decompositions of a braid as product of generators. Applications are given which are the motivations of the author: constructions of A. J. Bondal and M. M. Kapranov on exceptional systems [Math. USSR, Izv. 35, No. 3, 519-541 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1183-1205 (1989; Zbl 0703.14011)] about filtrations of triangulated categories and of M. Broué and J. Michel [Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, in “Finite reductive groups: related structures and representations” (ed. M. Cabanes), Proc. Int. Conf., Luminy, France, Oct. 1994, Progr. Math. 141, 73-139 (1997)] on correspondences on flag manifolds.