We will discuss the following theorems concerning colimits taken in the infinity-category of monoidal DG-categories. (No familiarity with infinity-categories will be required or assumed.) The affine Hecke category is the monoidal colimit of its finite type Hecke subcategories. The category of D-modules on the loop group is the monoidal colimit of the categories of D-modules on standard parahoric subgroups. These theorems, and many analogous ones, are consequences of a general inductive characterization of colimits indexed by a category of `words' in a Coxeter group. We will also discuss two applications-in-progress: the construction of a functor from the affine to the finite Hecke category in type A, and the construction of new `deformations' of the affine Hecke category.