Projective and anomalous representations of categories and their linearizations

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly $J\text{\hspace{0.17em}⁣}:\mathcal{C}\to 2\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}$ one can associate an extension ${\mathcal{C}}^J$ of $\mathcal{C}$ and a subcategory ${\mathcal{C}}_{\mathrm{S}\mathrm{T}}^J$ of ${\mathcal{C}}^J$ with the property that: (i) anomalous representations of $\mathcal{C}$ with anomaly $J$ are equivalent to $\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}$-linear functors $E\text{\hspace{0.17em}⁣}:{\mathcal{C}}^J\to \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}$, and (ii) these are in turn equivalent to linear representations of ${\mathcal{C}}_{\mathrm{S}\mathrm{T}}^J$ 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 $\alpha$, and linear representations of the central extension $G^{\alpha }$ of $G$ associated with $\alpha$.