A singular Coxeter presentation

We enlarge a Coxeter group into category, with one object for each finite parabolic subgroup, encoding the combinatorics of double cosets. This singular monoid, is connected to geometry partial flag varieties. Our main result presentation this category by generators and relations. also provide new description reduced expressions describe all braid relations between such expressions, prove an analogue Matsumoto's theorem. gives proper development ideas first introduced Geordie Williamson. In type A we equip monoid diagrammatic using webs.