Projective and anomalous representations of categories and their linearizations

We investigate the relation between projective and anomalous representations of categories, and show how to any anomaly J : C → 2 Vect one can associate an extension C^J of C and a subcategory C^J_ST of C^J with the property that: (i) anomalous representations of C with anomaly J are equivalent to Vect-linear functors E : C^J → Vect, and (ii) these are in turn equivalent to linear representations of C^J_ST where “J acts as scalars”. This construction, inspired by and generalizing the technique used to linearize anomalous functorial field theories in the physics literature, can be seen as a multi-object version of the classical relation between projective representations of a group G, with given 2-cocycle α, and linear representations of the central extension G^α of G associated with α.