**Abstract**

For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as *homological representations*. Representations of this kind have proved themselves especially important for the question of *linearity*, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear.

In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce *homological representation functors* encoding a large class of homological representations, defined on categories containing all *mapping class groups* and *motion groups* in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence-Bigelow, and yields many new representations.