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For groups of a topological origin, such as braid groups and mapping classgroups, an important source of interesting and highly non-trivialrepresentations is given by their actions on the twisted homology of associatedspaces; these are known as homological representations. Representations of thiskind have proved themselves especially important for the question of linearity,a key example being the family of topologically-defined representationsintroduced by Lawrence and Bigelow, and used by Bigelow and Krammer to provethat braid groups are linear. In this paper, we give a unified foundation forthe construction of homological representations using a functorial approach.Namely, we introduce homological representation functors encoding a large classof homological representations, defined on categories containing all mappingclass groups and motion groups in a fixed dimension. These source categoriesare defined using a topological enrichment of the Quillen bracket constructionapplied to categories of decorated manifolds. This approach unifies manypreviously-known constructions, including those of Lawrence-Bigelow, and yieldsmany new representations.

Topological representations of motion groups and mapping class groups -- a unified functorial construction