A chambered set for a Coxeter system is a set with an action by the Coxeter group and a specified set of orbit representatives, called the fundamental chamber, the stabilizer of each point of which is a standard parabolic subgroup. This notion arises naturally in describing dependence of facets of subarrangements of the Coxeter arrangement on the ambient space of the arrangement, which could be the Tits cone, Coxeter complex, Davis complex, imaginary cone, Cayley graph etc. This talk will discuss a monoidal category of chambered sets, and related structure. For finite Coxeter groups, part of this structure categorifies the descent algebra and certain modules for it, as studied by Solomon, Tits and Moszkowski. The results of Solomon and Tits for general Coxeter groups imply that the descent algebra has no satisfactory analogue, as an algebra, in general; the monoidal category of chambered sets provides a refined alternative.