Groups of a topological origin, such as braid groups and mapping class groups, often have "wild" representation theory. One would therefore like to construct representations of these groups topologically, in order to use topological tools to study them. A key example is Bigelow and Krammer's proof of the linearity of the braid groups, which uses topologically-defined representations due to Lawrence and Bigelow.
We give a unified construction of such topological representations: a large family of representations of a category containing all mapping class groups and motion groups in a fixed dimension. This unifies many previously-known constructions, including those of Lawrence-Bigelow, together with many new families of representations. Varying one parameter of the construction also produces pro-nilpotent towers of representations, of which we give several non-trivial low-dimensional examples. Moreover, the rich structure of this category helps to clarify the representation theory of these families of groups.