A good way to handle systems of braid group representations is to consider the category of functors $\mathcal{C}\to R\textrm{-Mod}$, where $\mathcal{C}$ is a category with the braid groups as automorphisms. The braid groupoid $\beta$ (ie the groupoid with natural numbers as objects and braid groups as automorphisms) is then a subcategory of such $\mathcal{C}$. Note that $\beta$ itself is not quite satisfactory for such $\mathcal{C}$ since a functor $\beta\to R\textrm{-Mod}$ encodes a family of representations where the representations of $B_{n}$ is independent of the one of $B_{n+1}$. In other words, we would like $\mathcal{C}$ to encode compatibilities between the representations. There already exist good candidates for such category:
The latter has the significant advantage that a large class of classical families of representations of the braid groups define functors $\mathcal{U}\beta\to R\textrm{-Mod}$:
the Burau representations; see Example 4.3 of [RWW].
the Tong-Yang-Ma and Lawrence-Krammer-Bigelow representations; see Section 1.2 of https://arxiv.org/pdf/1702.08279.pdf [S1].
the whole family of the Lawrence-Bigelow representations; see Section 5.2.1.1 of https://arxiv.org/pdf/1910.13423.pdf [PS].
Also, there are notions of polynomiality which allows us to characterise and prove more properties on these systems of representations:
the notion of (strong) polynomiality, a.k.a finite degree coefficient systems: the Burau representation is of degree $1$ (see Example 4.15 of [RWW]), the Tong-Yang-Ma representation is of degree $1$ and Lawrence-Krammer-Bigelow representations is of degree $2$ (see Propositions 3.25 and 3.33 of [S1]). This is the appropriate notion to prove twisted homological stability result see [RWW].
the notion of weak polynomial functors, which has originally been introduced for symmetric monoidal categories (for instance $FI$) by Djament and Vespa https://arxiv.org/abs/1308.4106, and generalised to categories of the same type as $\mathcal{U}\beta$ (namely pre-braided monoidal categories) in https://arxiv.org/pdf/1709.04278.pdf (see Section 4.2). An advantage of this notion is that it reflects more accurately than the strong polynomiality the behaviour of functors in the stable range. For instance, Church, Miller, Nagpal and Reinhold https://arxiv.org/pdf/1706.03845.pdf compute the weak polynomial degree (named "stable degree" in this paper) of some FI-modules. Moreover, denoting by $Pol_{d}(\mathcal{U}\beta)$ the category of weak polynomial functor of degree less or equal to $d$, we can define the quotient category
$$Pol_{d+1}(\mathcal{U}\beta)/Pol_{d}(\mathcal{U}\beta).$$
These quotient categories provide a new tool to handle families of representations with a sensible way to classify them. In particular, it doesn’t involve decomposition into irreducibles. See also Palmer https://arxiv.org/pdf/1712.06310.pdf for a comparison of the various instances of the notions of twisted coefficient system and polynomial functor. Hence weak polynomiality might be viewed as a refinement of representation stability phenomena and a sensible to talk about system of braid group representations.
