  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.
