embedded graph $W_n^S \subset S^{2n}$ by words in the blocks of $\lambda$. In other words, the basis is the set of tuples $\mathbf{w}$ of the form (2.7) if $S = \mathbb{T}$ and (2.8) if $S = \mathbb{M}$ (ignoring the initial tuple of words corresponding to the linear part of the graph, which does not exist in the mapping class group setting).

**Boundary connected sums.** We will use the following general principle for representations of the mapping class group of a surface that splits as a boundary connected sum. Let $\hat{\mathcal{L}}_{(\lambda,\ell)}$ be either $\mathcal{L}_{(\lambda,\ell)}^*$ or its vertical-type alternative $\mathcal{L}_{(\lambda,\ell)}^{*,v}$. We recall from §1.1.2.2 that each object $\mathbf{n}$ of $\mathcal{M}$ is the surface $S^{2n}$.

**Lemma 3.28** Let $\mathbf{n}, \mathbf{m} \in \text{Obj}(\mathcal{M})$. Let $\mathbf{w}$ be a basis element (see Notation 2.20) of the representation $\hat{\mathcal{L}}_{(\lambda,\ell)}(\mathbf{n}\mathbf{m})$, using the bases described in §2 and write the tuple $\mathbf{w}$ as $\mathbf{w} = (\mathbf{w}', \mathbf{w}'')$, where the entries of $\mathbf{w}'$ correspond to arcs supported in $S^{2n}$ and the entries of $\mathbf{w}''$ correspond to arcs supported in $S^{2m}$. Then, for $\varphi_{\mathbf{m}} \in \text{MCG}(S^{2m})$, the element $\hat{\mathcal{L}}_{(\lambda,\ell)}(\text{id}_{\mathbf{n}\mathbf{m}}\varphi_{\mathbf{m}})(\mathbf{w})$ is a linear combination of basis elements of the form $(\mathbf{w}', \mathbf{v}'')$, where $\mathbf{v}''$ runs over all possible labellings of arcs supported in $S^{2m}$.

*Proof.* As in the proof of Lemma 3.15, we prefer to denote by $e_{\mathbf{w}}$ (rather than $\mathbf{w}$) the basis element corresponding to the tuple $\mathbf{w}$ for the sake of clarity. Let $e_{\mathbf{v}}'$ be an arbitrary dual basis element and write $\mathbf{v} = (\mathbf{v}', \mathbf{v}'')$ similarly to the decomposition $\mathbf{w} = (\mathbf{w}', \mathbf{w}'')$. It suffices to show that $\langle \hat{\mathcal{L}}_{(\lambda,\ell)}(\text{id}_{\mathbf{n}\mathbf{m}}\varphi_{\mathbf{m}})(e_{\mathbf{w}}), e_{\mathbf{v}}' \rangle = 0$ unless $\mathbf{v}' = \mathbf{w}'$. To see this, recall that the homology class $e_{\mathbf{w}}$ is represented by some configurations on embedded arcs in $S^{2(n+m)}$. Since $\text{id}_{\mathbf{n}\mathbf{m}}\varphi_{\mathbf{m}}$, by construction, is supported in $S^{2m}$, the homology class $\hat{\mathcal{L}}_{(\lambda,\ell)}(\text{id}_{\mathbf{n}\mathbf{m}}\varphi_{\mathbf{m}})(e_{\mathbf{w}})$ may be represented by some configurations on embedded arcs that are identical on the boundary connected summand $S^{2n}$ to those representing $e_{\mathbf{w}}$. The intersection pairing with $e_{\mathbf{v}}'$ must therefore be zero unless $\mathbf{v}' = \mathbf{w}'$. $\square$

**Interaction with the braiding.** To discuss elements of mapping class groups that act by “braiding” handles or crosscaps of the surface $\mathbf{n} = S^{2n}$, it is convenient to pass, in this section, to a different way of representing $S^{2n}$ diagrammatically. Instead of a rectangle to which we have glued a finite number of strips (as in, for example, Figure 2.1), we will represent this surface as a rectangle from which we have either erased the interiors of 2g discs and glued their boundaries in pairs (when considering $S^{2g} \cong \Sigma_{g,1}$) or erased the interiors of h discs and glued each resulting boundary component to itself by a degree-2 map (when considering $S^{2h} \cong \mathcal{N}_{h,1}$).

Each basis element $\mathbf{w}$ of the representation $\mathcal{L}_{(\lambda,\ell)}^*(\mathbf{n})$ (see Figures 2.2c and 2.2d) looks as illustrated in Figure 3.5 in this picture, where we have also included explicit choices of tethers (see §2.4), i.e. paths from the base configuration to a point on the submanifold representing the homology class. Similarly, we denote by $\mathbf{w}^v$ the basis elements of the vertical-type alternative representations $\mathcal{L}_{(\lambda,\ell)}^{*,v}(\mathbf{n})$ (see Figures 2.3c and 2.3d), which look as illustrated in Figure 3.6 in this picture, where again we have included explicit choices of tethers.

**Notation 3.29** We denote by $\sigma_1 \in \text{MCG}(S^{2n})$ the mapping class illustrated in Figure 3.7: it braids the left-most two handles if $S = \mathbb{T}$ and it braids the left-most two crosscaps if $S = \mathbb{M}$.

**Lemma 3.30** The following identities hold in $\mathcal{L}_{(\lambda,\ell)}^*(\mathbf{n})$ for $\ell \ge 1$:

$$ \sigma_1^{-1}(([\emptyset, \emptyset], [w_3, w_4], \dots)) = ([w_3, w_4], [\emptyset, \emptyset], \dots) \quad (3.19) $$

$$ \sigma_1^{-1}(([\emptyset], [w_2], \dots)) = ([w_2], [\emptyset], \dots) \quad (3.20) $$

and in $\mathcal{L}_{(\lambda,\ell')}^v(\mathbf{n})$ for $\ell' \le 2$ (recall that $\mathcal{L}_{(\lambda,\ell')}^v = \mathcal{L}_{(\lambda,\ell')}^{u,v}$ by Lemma 1.13):

$$ \sigma_1 (([\emptyset, \emptyset], [w_3, w_4], \dots)^v) = ([w_3, w_4], [\emptyset, \emptyset], \dots)^v \quad (3.21) $$

$$ \sigma_1 (([\emptyset], [w_2], \dots)^v) = ([w_2], [\emptyset], \dots)^v \quad (3.22) $$

*Proof.* Equations (3.19) and (3.20) are clear from the diagrams of the basis element (see Figure 3.5) and the action of $\sigma_1$ (see Figure 3.7), using the facts that the label(s) corresponding to the left-most handle or crosscap are empty and that the tethers are the same up to homotopy.

On the other hand, we see from the diagrams that the left-hand side of equation (3.21) is equal to the element illustrated in Figure 3.8a. This differs from the right-hand side of equation (3.21)
<figure><img src="image_14.png" /><figcaption>(a) The orientable case.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) The non-orientable case.</figcaption></figure>

Figure 3.5 Another perspective on the basis elements depicted in Figures 2.2c and 2.2d.

<figure><img src="image_16.png" /><figcaption>(a) The orientable case.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) The non-orientable case.</figcaption></figure>

Figure 3.6 Another perspective on the basis elements depicted in Figures 2.3c and 2.3d.
<figure><img src="image_18.png" /><figcaption>Figure 3.7 The braiding element $\sigma_1 \in \text{MCG}(S)$ when $S = \Sigma_{g,1}$ (left) and $S = N_{h,1}$ (right).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 3.8 The left-hand side of equation (3.21) differs from the right-hand side of equation (3.21) by the scalar in $\mathbb{Z}[Q_{(\lambda,2)}(\mathbb{T})]$ given by the image in $Q_{(\lambda,2)}(\mathbb{T})$ of the loop illustrated on the right.</figcaption></figure>

only in the choice of tether. As explained in Lemma 2.22, changing the tether has the effect of multiplying the homology class by the unit in the ground ring given by the monodromy action of the based loop of configurations given by the difference between the two tethers. In our setting, the monodromy action is the quotient onto $Q_{(\lambda,\ell')}(\mathbb{T})$ followed by its inclusion into $\mathbb{Z}[Q_{(\lambda,\ell')}(\mathbb{T})]^\times$. We therefore need to show that this based loop of configurations, which is illustrated in Figure 3.8b, projects to the trivial element of the group $Q_{(\lambda,\ell')}(\mathbb{T})$. This is obvious for $\ell' = 1$ since the group $Q_{(\lambda,1)}(\mathbb{T})$ is always trivial. For $\ell' = 2$, we recall from Lemma 1.21 that $Q_{(\lambda,2)}(\mathbb{T})$ is simply a product of copies of $\mathbb{Z}/2$, one for each block of $\lambda = (\lambda_1, \dots, \lambda_r)$ with $\lambda_i \ge 2$. The projection onto $Q_{(\lambda,2)}(\mathbb{T})$ records the writhe (modulo 2) of each block of strands (in a surface of positive genus the writhe is only well-defined modulo 2). It is clear that the writhe of the loop of configurations illustrated in Figure 3.8b is trivial for each block. This establishes equation (3.21).

We argue similarly for equation (3.22). (Again, the case $\ell' = 1$ being obvious, we just consider $\ell' = 2$.) The left-hand side is equal to the element illustrated in Figure 3.9a, which differs from the right-hand side only by its choice of tether; the difference between the two tethers forms the based loop of configurations illustrated in Figure 3.9b. We therefore just have to show that this projects to the trivial element of the group $Q_{(\lambda,2)}(\mathbb{M})$. This time the group $Q_{(\lambda,2)}(\mathbb{M})$ is a product of $r' + r$ copies of $\mathbb{Z}/2$, where $r'$ denotes the number of blocks of $\lambda$ with $\lambda_i \ge 2$; see Lemma 1.21. The first $r'$ copies of $\mathbb{Z}/2$, in the projection to $Q_{(\lambda,2)}(\mathbb{M})$ of a loop of configurations, record the writhe of each block of strands; see [PS24a, Prop. 1.1]. The remaining $r$ copies of $\mathbb{Z}/2$ record, for each block of strands, the number of times modulo 2 that a strand from that block passes through a crosscap; [PS24a, Prop. 1.1]. As before, it is clear that the writhe of the loop of configurations illustrated in Figure 3.9b is trivial for each block; thus the first $r'$ coordinates of its projection to $Q_{(\lambda,2)}(\mathbb{M})$ are zero. Moreover, each strand in this loop of configurations passes around a crosscap an integer number of times, which corresponds to passing through a crosscap an even number of times; thus the last $r$ coordinates of its projection to $Q_{(\lambda,2)}(\mathbb{M})$ are also zero. This establishes equation (3.22). $\square$

**Remark 3.31** Equations (3.19) and (3.20) of Lemma 3.30 hold for all $\ell \ge 1$. On the other hand, we used the explicit structure of the quotient group $Q_{(\lambda,2)}(\mathbb{S})$ (and the fact that $Q_{(\lambda,1)}(\mathbb{S})$ is trivial) to prove equations (3.21) and (3.22). For $\ell' \ge 3$ the proof shows that these equations hold up to a unit scalar, which is the image in $Q_{(\lambda,\ell')}(\mathbb{S})$ of the loops in Figures 3.8b and 3.9b for $\mathbb{S} = \mathbb{T}$ and $\mathbb{S} = \mathbb{M}$ respectively. We do not know whether this scalar is trivial in these cases.
<figure><img src="image_20.png" /><figcaption>Figure 3.9 The left-hand side of equation (3.22) differs from the right-hand side of equation (3.22) by the scalar in $\mathbb{Z}[Q_{(\lambda,2)}(\mathbb{M})]$ given by the image in $Q_{(\lambda,2)}(\mathbb{M})$ of the loop illustrated on the right.</figcaption></figure>

**Remark 3.32** We will view equations (3.21) and (3.22) as being statements about the action of $(\sigma_1^{-1})^\dagger = \sigma_1$, where $(-)^\dagger$ is the operation that inverts the braiding of a braided monoidal category; see the beginning of §1.1.

**Preliminary study of the difference functor.** Similarly to $\mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n})$, the MCG($S^{\sharp n}$)-module $\tau_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n}) = H_k^{\mathrm{BM}}(C_\lambda(S^{\sharp(1+n)} \setminus I); \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)])$ is a free $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module with basis indexed by labellings of the embedded graph $\mathbb{W}_{1+n}^S \subset S^{\sharp(1+n)}$ by words in the blocks of $\lambda$. Recall from Lemma 1.11 that the image of the morphism $[1, \mathrm{id}_{1\sharp n}] : \mathbf{n} \to 1\sharp n$ of $\langle \mathcal{M}, \mathcal{M} \rangle$ under $\mathcal{L}^*_{(\lambda,\ell)}$ is the map $H_k^{\mathrm{BM}}(C_\lambda(S^{\sharp n} \setminus I); \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]) \to H_k^{\mathrm{BM}}(C_\lambda(S^{\sharp(1+n)} \setminus I); \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)])$ induced by the inclusion of configuration spaces $C_\lambda(S^{\sharp n} \setminus I) \hookrightarrow C_\lambda(S^{\sharp(1+n)} \setminus I)$ coming from the boundary connected sum with the left-most copy of $S$. It thus follows that the map $\mathcal{L}^*_{(\lambda,\ell)}([1, \mathrm{id}_{1\sharp n}])$ is the injection defined on basis elements by adjoining the empty word as the label of the left-most edge, or edges, of $\mathbb{W}_{1+n}^S$, explicitly given by

$$ \left\{ \begin{array}{ll} ([w_1, w_2], \dots, [w_{2n-1}, w_{2n}]) \mapsto ([\emptyset, \emptyset], [w_1, w_2], \dots, [w_{2n-1}, w_{2n}]) & \text{if } S = \mathbb{T}; \\ ([w_1], \dots, [w_n]) \mapsto ([\emptyset], [w_1], \dots, [w_n]) & \text{if } S = \mathbb{M}. \end{array} \right. \qquad (3.23) $$

Hence $\kappa_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n}) = 0$ and $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n})$ is the free $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module with generating set given by the tuples $([w_0, w'_0], [w_1, w_2], \dots, [w_{2n-1}, w_{2n}])$ such that $|w_0| + |w'_0| \ge 1$ if $S = \mathbb{T}$, and the tuples $([w_0], [w_1], \dots, [w_n])$ such that $|w_0| \ge 1$ if $S = \mathbb{M}$. Then, we define a $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module injection

$$ (\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n}) : \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n}) \hookrightarrow \tau_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n}) \qquad (3.24) $$

by sending each generating tuple $\mathbf{w} = ([w_0], [w_1], \dots, [w_n])$ of $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n})$ to itself in $\tau_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n})$.

**Difference functor decomposition.** From now on in §3.3.1, we assume that $k \ge 2$. In order to identify $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{n})$ in terms of some $\mathcal{L}^*_{(\lambda',\ell)}(\mathbf{n})$ where $\lambda' < \lambda$, we define some $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module morphisms depending on $S$ as follows.

For $S = \mathbb{T}$. For each object $\mathbf{n} \ge 0$, we have to define several injections into $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\Gamma)(\mathbf{n})$. First, for each fixed pair of positive integers $(j_1, j_2)$ such that $1 \le j_1, j_2 \le r$, we define an injection $\tau_1 \mathcal{L}^*_{(\lambda[j_1, j_2], \ell)}(\Gamma)(\mathbf{n}) \hookrightarrow \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\Gamma)(\mathbf{n})$ by mapping each element $([w_1, w_2], \dots, [w_{2n+1}, w_{2n+2}])$ to $([j_1 w_1, j_2 w_2], \dots, [w_{2n+1}, w_{2n+2}])$. We denote by $(i_{(\lambda,\ell)}(\Gamma))_{\mathbf{n}}^{\{\lambda-2\}}$ the direct sum over $1 \le j_1, j_2 \le r$ of these injections.

Furthermore, we introduce two subfunctors of $\tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)$ (see Proposition 3.33), in order to define some more injections into $\delta_1 \mathcal{L}^*_{(\lambda, \ell)}(\Gamma)(\mathbf{n})$. For each object $\mathbf{m} \ge 0$, let $\tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)_{[-, \emptyset]}(\mathbf{m})$ and $\tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)_{[\emptyset, -]}(\mathbf{m})$ be the free $\mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{T})]$-modules with basis all the generators of $\tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)(\mathbf{m})$ of the form $([w_1, \emptyset], \dots, [w_{2m+1}, w_{2m+2}])$ and $([\emptyset, w_2], \dots, [w_{2m+1}, w_{2m+2}])$ respectively. These modules are direct summands of $\tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)(\mathbf{m})$.

**Proposition 3.33** By gathering the $\mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathbb{T})]$-modules $\{\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\Gamma)_{[-,\emptyset]}(\mathbf{m})\}_{\mathbf{m} \in \mathrm{Obj}(\mathcal{M}_2^+)}$ and $\{\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\Gamma)_{[\emptyset,-]}(\mathbf{m})\}_{\mathbf{m} \in \mathrm{Obj}(\mathcal{M}_2^+)}$ respectively, and with the assignment of the functor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\Gamma)$ on the morphisms of $\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$, we define functors $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\Gamma)_{[-,\emptyset]}$ and $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\Gamma)_{[\emptyset,-]}$ of the
category Fct($\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$, $\mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathbb{T})]$)-Mod$^\bullet$). Moreover, these are direct summands of the functor
$\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})$.

Proof. We focus here on the proof for $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}$, that for $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[\emptyset,-]}$ being a mutatis mutandis replication. For each $\mathfrak{m} \ge 0$, we denote by $\mathbf{i}_{[-,\emptyset](\mathfrak{m})}$ (resp. $\mathbf{p}_{[-,\emptyset](\mathfrak{m})}$) the canonical injection (resp. projection) associated to the direct summand $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset](\mathfrak{m})}$ of $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})(\mathfrak{m})$. For all $\varphi \in \Gamma_{m,1}$, it follows from Lemma 3.28 that $\mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma}) (\mathrm{id}_1 \otimes \varphi)$ does not affect the first two entries of each generator ($[w_1, \emptyset], \dots, [w_{2m+1}, w_{2m+2}]$), so the image of the composite $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})(\varphi) \circ \mathbf{i}_{[-,\emptyset](\mathfrak{m})}$ belongs to the module $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset](\mathfrak{m})}$. We deduce that the $\mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathbb{T})]$-modules $\{\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset](\mathfrak{m})}\}_{\mathfrak{m} \in \mathrm{Obj}(\mathcal{M}_2^+)}$ define a functor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]} : \mathcal{M}_2^+ \to \mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathbb{T})]$-Mod$^\bullet$ by assigning the $\Gamma_{m,1}$-module structure of $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})(\mathfrak{m})$ for each $\mathfrak{m} \ge 0$.

We now extend the functor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}$ along the inclusion $\mathcal{M}_2^+ \hookrightarrow \langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$. Recall from (1.2) that $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{1\mathfrak{m}}]) = \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})(\sigma_1^{-1}) \circ \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{2\mathfrak{m}}])$, where $\sigma_1 = b_{1,1}^{\mathcal{M}_2^+} \otimes \mathrm{id}_\mathfrak{m}$ and $b_{1,1}^{\mathcal{M}_2^+}$ is the braiding of $\mathcal{M}_2^+$ (see Figure 3.7). Note that $\mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{2\mathfrak{m}}])$ has a similar description to that of $\mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{1\mathfrak{m}}])$, by simply replacing $n$ with $n+1$ in the defining assignment (3.23). Then, we deduce that $(\mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{2\mathfrak{m}}]))([w_1, \emptyset], \dots, [w_{2m+1}, w_{2m+2}])$ is equal to $([\emptyset, \emptyset], [w_1, \emptyset], \dots, [w_{2m+1}, w_{2m+2}])$ for each element of $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}(\mathfrak{m})$. So it follows from equation (3.19) of Lemma 3.30 that $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})([1, \mathrm{id}_{1\mathfrak{m}}])([w_1, \emptyset], \dots, [w_{2m+1}, w_{2m+2}]) = ([w_1, \emptyset], [\emptyset, \emptyset], \dots, [w_{2m+1}, w_{2m+2}])$, and thus belongs to $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}([1\mathfrak{m}])$. Relation (1.3) is then automatically satisfied since $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})$ is a functor out of $\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$; it thus follows from Lemma 1.1 that $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}$ extends to an object of Fct($\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$, $\mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathbb{T})]$)-Mod$^\bullet$).

Finally, a straightforward adaptation of the above reasoning proves that all of the projection maps $\{\mathbf{p}_{[-,\emptyset](\mathfrak{m})}\}_{\mathfrak{m} \in \mathrm{Obj}(\mathcal{M}_2^+)}$ are equivariant with respect to all of the morphisms of the category $\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$. Therefore, the subfunctor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}$ is a direct summand of the functor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})$. $\square$

Now, for each $1 \le j \le r$, we define an injection $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]}(\mathfrak{n}) \hookrightarrow \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{\Gamma})(\mathfrak{n})$ by mapping each element $([w_1, \emptyset], \dots, [w_{2n+1}, w_{2n+2}])$ to $([jw_1, \emptyset], \dots, [w_{2n+1}, w_{2n+2}])$, and an injection $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[\emptyset,-]}(\mathfrak{n}) \hookrightarrow \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathbf{\Gamma})(\mathfrak{n})$ by mapping each element $([\emptyset, w_2], \dots, [w_{2n+1}, w_{2n+2}])$ to $([\emptyset, jw_2], \dots, [w_{2n+1}, w_{2n+2}])$. We denote by $(i_{(\lambda,\ell)}(\mathbf{\Gamma}))_{\mathfrak{n}}^{\{\lambda-1\}}$ the direct sum over $1 \le j \le r$ of these two injections, and by $(i_{(\lambda,\ell)}(\mathbf{\Gamma}))_{\mathfrak{n}}^{\{\lambda-1\}}$ and $(i_{(\lambda,\ell)}(\mathbf{\Gamma}))_{\mathfrak{n}}^{\{\lambda-2\}}$. Finally, we use the notation $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[\emptyset]}$ for the subfunctor $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[-,\emptyset]} \oplus \tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})_{[\emptyset,-]}$ of $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathbf{\Gamma})$.

For $S = \mathbb{M}$. For each $1 \le j \le r$ and each $n \ge 0$, we define an injection $\tau_1 \mathcal{L}^*_{(\lambda[j],\ell)}(\mathcal{N})(\mathfrak{n}) \hookrightarrow \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathcal{N})(\mathfrak{n})$ by mapping each element $([w_1], \dots, [w_{n+1}])$ to $([jw_1], \dots, [w_{n+1}])$. We denote by $(i_{(\lambda,\ell)}(\mathcal{N}))_n$ the direct sum over $1 \le j \le r$ of these injections.

**Notation 3.34** To abbreviate and unify our notation, we set $(i_{(\lambda,\ell)})_n$ to be the map $(i_{(\lambda,\ell)}(\mathbf{\Gamma}))_n$ when $S = \mathbb{T}$, and the map $(i_{(\lambda,\ell)}(\mathcal{N}))_n$ when $S = \mathbb{M}$. We also denote by $\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda',\ell)}$ the functor $(\bigoplus_{1 \le j_1, j_2 \le r} \tau_1 \mathcal{L}^*_{(\lambda[j_1, j_2], \ell)}(\mathbf{\Gamma})) \oplus (\bigoplus_{1 \le j \le r} \tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\mathbf{\Gamma})_{[\emptyset]})$ for the case when $S = \mathbb{T}$, and the functor $\bigoplus_{1 \le j \le r} \tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\mathcal{N})$ for the case when $S = \mathbb{M}$.

An elementary check of the bases of $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathfrak{n})$ and $\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda',\ell)}(\mathfrak{n})$ shows that the morphism $(i_{(\lambda,\ell)})_n$ is an isomorphism of (free) $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-modules for each $n \ge 0$:

$$ (i_{(\lambda,\ell)})_n : \bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda',\ell)}(\mathfrak{n}) \xrightarrow{\cong} \delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathfrak{n}). \quad (3.25) $$

A similar study to the above holds in the vertical setting. Namely, Proposition 2.9 provides $\mathcal{L}^v_{(\lambda,\ell)}(\mathfrak{n})$ with the analogous free $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module structures, indexed by the set of tuples labelling the embedded “vertical” graphs modelled by Figures 2.3c and 2.3d. The above reasoning then repeats mutatis mutandis to provide an analogous isomorphism $(i^v_{(\lambda,\ell)})_n$ to (3.25).
### 3.3.2. Classical homological representation functors

We prove here Theorem A for the homological representation functors of mapping class groups in the classical (non-vertical) setting. Following our notation so far in §3.3, we make all further reasoning, as much as possible, on the (generic) functor $\mathcal{L}^*_{(\lambda, \ell)}$, which denotes any one of the functors (1.17) and (1.18).

**Theorem 3.35** For each ordered partition $\lambda \vdash k \ge 2$, each integer $\ell \ge 1$ and $\star$ either the blank space or $\star = u$, the exact sequence (3.1) induces the following isomorphisms in the functor categories $\mathrm{Fct}(\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle, \mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{T})]$-Mod$^\bullet$) and $\mathrm{Fct}(\langle \mathcal{M}_2^-, \mathcal{M}_2^- \rangle, \mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{M})]$-Mod$^\bullet$) respectively:

$$ \tau_1 \mathcal{L}^*_{(\lambda, \ell)}(\Gamma) \cong \mathcal{L}^*_{(\lambda, \ell)}(\Gamma) \oplus \left( \bigoplus_{1 \le j_1, j_2 \le r} \tau_1 \mathcal{L}^*_{(\lambda[j_1, j_2], \ell)}(\Gamma) \right) \oplus \left( \bigoplus_{1 \le j \le r} \tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\Gamma)[\emptyset] \right), \quad (3.26) $$

$$ \tau_1 \mathcal{L}^*_{(\lambda, \ell)}(\mathcal{N}) \cong \mathcal{L}^*_{(\lambda, \ell)}(\mathcal{N}) \oplus \left( \bigoplus_{1 \le j \le r} \tau_1 \mathcal{L}^*_{(\lambda[j], \ell)}(\mathcal{N}) \right). \quad (3.27) $$

*Proof.* As a consequence of the above preliminary study of §3.3.1, it suffices to show that the $\mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{S})]$-module isomorphisms $\{(i_{(\lambda, \ell)})_n\}_{n \in \mathrm{Obj}(\mathcal{M})}$ (see (3.25)) assemble into an isomorphism $i_{(\lambda, \ell)}: \bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda', \ell)} \xrightarrow{\sim} \delta_1 \mathcal{L}^*_{(\lambda, \ell)}$ in $\mathrm{Fct}(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{S})]$-Mod$^\bullet$), while the $\mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{S})]$-module injections $\{(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n\}_{n \in \mathrm{Obj}(\mathcal{M})}$ (see (3.24)) assemble into a natural transformation that is a section of $\Delta_1 \mathcal{L}^*_{(\lambda, \ell)}$.

First, we prove the commutation of $(i_{(\lambda, \ell)})_n$ and of $(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n$ with respect to the action of MCG($S^{\mathfrak{h}_n}$). For each element $\rho$ of MCG($S^{\mathfrak{h}_n}$) and for each direct summand $\tau_1 \mathcal{L}^*_{(\lambda', \ell)}$ of the source of $(i_{(\lambda, \ell)})_n$ (see (3.25)), the morphisms $\delta_1 \mathcal{L}^*_{(\lambda, \ell)}(\rho)$ and $\tau_1 \mathcal{L}^*_{(\lambda', \ell)}(\rho)$ are induced by the actions of $\mathcal{L}^*_{(\lambda, \ell)}(\mathrm{id}_1 \mathfrak{h} \rho)$ and of $\mathcal{L}^*_{(\lambda', \ell)}(\mathrm{id}_1 \mathfrak{h} \rho)$ respectively on the Borel-Moore homology classes supported on the embedded graph $W_{1+n}^S \subset S^{\mathfrak{h}(1+n)}$. It follows from Lemma 3.28 that the actions of $\mathcal{L}^*_{(\lambda, \ell)}(\mathrm{id}_1 \mathfrak{h} \rho)$ and of $\mathcal{L}^*_{(\lambda', \ell)}(\mathrm{id}_1 \mathfrak{h} \rho)$ do not affect the first two (if $S = \mathbb{T}$) or one (if $S = \mathbb{M}$) entries of a tuple corresponding to a generator. Since $(i_{(\lambda, \ell)})_n$ and $(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n$ both only affect the first two (if $S = \mathbb{T}$) or one (if $S = \mathbb{M}$) entries of a tuple by their definitions, we deduce that $\delta_1 \mathcal{L}^*_{(\lambda, \ell)}(\rho) \circ (i_{(\lambda, \ell)})_n = (i_{(\lambda, \ell)})_n \circ \bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda', \ell)}(\rho)$ and $(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n \circ \delta_1 \mathcal{L}^*_{(\lambda, \ell)}(\rho) = \tau_1 \mathcal{L}^*_{(\lambda, \ell)}(\rho) \circ (\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n$. Therefore, the morphisms $\{(i_{(\lambda, \ell)})_n\}_{n \in \mathrm{Obj}(\mathcal{M})}$ and $\{(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n\}_{n \in \mathrm{Obj}(\mathcal{M})}$ define natural transformations $i_{(\lambda, \ell)}$ and $\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)}$ in $\mathrm{Fct}(\mathcal{M}, \mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{S})]$-Mod$^\bullet$).

Now, by Lemma 1.2, it suffices to check relation (1.4) in order to prove that $i_{(\lambda, \ell)}$ and $\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)}$ are actually natural transformations of functors on $\langle \mathcal{M}, \mathcal{M} \rangle$. We consider an object $n \ge 1$, the proof being trivial for $n = 0$. We note from the formula (1.2), from the composition rule (1.1), from the fact that $\mathfrak{h}$ is a strict monoidal structure and from the functoriality of $\mathcal{L}^*_{(\lambda, \ell)}$, that $\tau_1 \mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}]) = \mathcal{L}^*_{(\lambda, \ell)}(\sigma_1^{-1}) \circ \mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{2\mathfrak{h}_n}])$, where $\sigma_1 \in \mathrm{Aut}_{\mathcal{M}}(S^\mathfrak{h}S^\mathfrak{h}S^n)$ is the element $b_{1,1}^M \mathfrak{h} \mathrm{id}_n$ defined by the braiding $b_{1,1}^M: \mathfrak{h}1 \cong \mathfrak{h}1$ (see Figure 3.7). Note that $\mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{2\mathfrak{h}_n}])$ has a similar description to that of $\mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}])$, by simply replacing $n$ with $n+1$ in the defining assignment (3.23). More precisely, it is the map induced by the embedding of $W_{1+n}^S$ into $W_{2+n}^S$ given by sending the $i$th edge ($S^1 - \mathrm{pt})_i$ to the $(i+2)$nd edge ($S^1 - \mathrm{pt})_{i+2}$ (if $S = \mathbb{T}$) or the $(i+1)$st edge ($S^1 - \mathrm{pt})_{i+1}$ (if $S = \mathbb{M}$). In particular this implies that, in the image of $\mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{2\mathfrak{h}_n}])$, there are no configuration points on the two first edges ($S^1 - \mathrm{pt})_1$ and ($S^1 - \mathrm{pt})_2$ if $S = \mathbb{T}$ or on the first edge ($S^1 - \mathrm{pt})_1$ if $S = \mathbb{M}$. Then the morphism $\mathcal{L}^*_{(\lambda, \ell)}(\sigma_1^{-1})$ corresponds to the action of $\sigma_1^{-1}$ on $W_{2+n}^S$. It therefore follows from equations (3.19) and (3.20) of Lemma 3.30 that for any generator $\mathbf{w}$ of $\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda', \ell)}(\mathbf{n})$, both $\delta_1 \mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}])((i_{(\lambda, \ell)})_n(\mathbf{w}))$ and $(i_{(\lambda, \ell)})_{1+n}(\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda', \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}])(\mathbf{w}))$ are equal to $\mathfrak{o}((i_{(\lambda, \ell)})_n(\mathbf{w}))$, where $\mathfrak{o}$ denotes the operation

$$ \left\{ \begin{array}{ll} ([w_1, w_2], [w_3, w_4], \dots) & \mapsto ([w_1, w_2], [\emptyset, \emptyset], [w_3, w_4], \dots) & \text{if } S = \mathbb{T} \\ ([w_1], [w_2], \dots) & \mapsto ([w_1], [\emptyset], [w_2], \dots) & \text{if } S = \mathbb{M}. \end{array} \right. $$

The same arguments using Lemma 3.30 also prove that $\tau_1 \mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}])((\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_n(\mathbf{w}'))$ and $(\Delta'_1 \mathcal{L}^*_{(\lambda, \ell)})_{1+n}(\delta_1 \mathcal{L}^*_{(\lambda, \ell)}([1, \mathrm{id}_{1\mathfrak{h}_n}])(\mathbf{w}'))$ are both equal to $\mathfrak{o}(\mathbf{w}')$ for any generator $\mathbf{w}'$ of $\delta_1 \mathcal{L}^*_{(\lambda, \ell)}(\mathbf{n})$.
It then follows from the above equalities and induction on $\mathfrak{m} \ge 1$ that the collections of morphisms $\{(i_{(\lambda,\ell)})_n\}_{n \in \text{Obj}(\mathcal{M})}$ and $\{(\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)})_n\}_{n \in \text{Obj}(\mathcal{M})}$ commute with the action of $[\mathfrak{m}, \text{id}_{\mathfrak{m}\otimes\mathfrak{n}}]$ for each $\mathfrak{m} \ge 1$. Hence relation (1.4) is satisfied for all $\mathfrak{n} \in \text{Obj}(\mathcal{M})$ and Lemma 1.2 implies that $i_{(\lambda,\ell)}$ and $\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)}$ are natural transformations of functors out of $\langle \mathcal{M}, \mathcal{M} \rangle$. In particular, we deduce that $i_{(\lambda,\ell)}$ is an isomorphism in $\text{Fct}(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathfrak{s})]\text{-Mod}^\bullet)$ between the functors $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}$ and $\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda',\ell)}$. Also, it follows from the definition of $(\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)})_n$ (see (3.24)) that $(\Delta_1 \mathcal{L}^*_{(\lambda,\ell)})_n \circ (\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)})_n = \text{id}_{\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(\mathfrak{n})}$, and so $\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)}$ is a section of $\Delta_1 \mathcal{L}^*_{(\lambda,\ell)}$ in $\text{Fct}(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathfrak{s})]\text{-Mod}^\bullet)$. Since $\kappa_1 \mathcal{L}^*_{(\lambda,\ell)} = 0$ (because $i_1(\mathcal{L}^*_{(\lambda,\ell)})_n$ is clearly injective for each $\mathfrak{n} \in \mathcal{M}$; see (3.23)), we deduce that the exact sequence (3.1) is a split short exact sequence, which provides the isomorphisms (3.26) and (3.27).

In the above arguments, whenever we deal with a twisted homological representation functor, we stress that the action on the ground ring $\mathbb{Z}[Q^*_{(\lambda,\ell)}(\mathfrak{s})]$ does not affect any of the above reasoning, since $\mathcal{L}^*_{(\lambda,\ell)}$ and $\bigoplus_{\lambda'} \tau_1 \mathcal{L}^*_{(\lambda',\ell)}$ are equipped with the same action as $\tau_1 \mathcal{L}^*_{(\lambda,\ell)}$ via the change of rings operation of Convention 3.8; see Lemma 3.7. $\square$

### 3.3.3. Vertical-type alternatives

We now deal with the vertical-type alternatives of the homological representation functors for the mapping class groups of surfaces introduced in §1.3.3. We consider the functors $\mathcal{L}^v_{(\lambda,\ell')}(\Gamma)$ for orientable surfaces and the functors $\mathcal{L}^v_{(\lambda,\ell')}(\mathcal{N})$ for non-orientable surfaces, only for $\ell' \le 2$. (We recall from Lemma 1.13 that $\mathcal{L}^v_{(\lambda,\ell')} = \mathcal{L}^{u,v}_{(\lambda,\ell')}$ in this case.) In particular, we do not consider the functors $\mathcal{L}^v_{(\lambda,\ell)}(\mathcal{N})$ with $\ell \ge 3$ for non-orientable surfaces. This is because the proof of Theorem 3.36 below relies on the identities proven in Lemma 3.30 using the specific structure of the transformation groups $Q_{(\lambda,2)}(\mathbb{T})$ and $Q_{(\lambda,2)}(\mathcal{M})$ (see Lemma 1.21); in contrast, the groups $Q_{(\lambda,\ell)}(\mathcal{M})$ are not known for $\ell \ge 3$. We however conjecture that all of the following arguments, including the results of Theorem 3.36 and Theorem C, also hold for $\ell \ge 3$.

**Theorem 3.36** For each ordered partition $\lambda \vdash k \ge 2$, the exact sequence (3.1) induces the analogous isomorphisms to (3.26) and (3.27) for the functors $\mathcal{L}^v_{(\lambda,1)}(\Gamma)$, $\mathcal{L}^v_{(\lambda,2)}(\Gamma)$, $\mathcal{L}^v_{(\lambda,1)}(\mathcal{N})$ and $\mathcal{L}^v_{(\lambda,2)}(\mathcal{N})$.

*Proof.* We fix $\ell' \in \{1,2\}$. The analogous isomorphisms to (3.26) and (3.27) for the vertical-type alternatives follow mutatis mutandis from the proof of Theorem 3.35 by defining analogues $(i^v_{(\lambda,\ell')})_n$ and $(\Delta'_1 \mathcal{L}^v_{(\lambda,\ell')})_n$ to the morphisms $(i_{(\lambda,\ell')})_n$ and $(\Delta'_1 \mathcal{L}^*_{(\lambda,\ell')})_n$ for each $\mathfrak{n} \in \text{Obj}(\mathcal{M})$. The proof that these define natural transformations $i^v_{(\lambda,\ell')}$ and $\Delta'_1 \mathcal{L}^v_{(\lambda,\ell')}$ in $\text{Fct}(\mathcal{M}, \mathbb{Z}[Q_{(\lambda,\ell')}]\text{-Mod})$ is a verbatim repetition of the first part of the proof of Theorem 3.35, again using the disjoint support argument of Lemma 3.28. Then, the proof that $i^v_{(\lambda,\ell')}$ and $\Delta'_1 \mathcal{L}^v_{(\lambda,\ell')}$ are natural transformations in $\text{Fct}(\langle \mathcal{M}^\dagger, \mathcal{M}^\dagger \rangle, \mathbb{Z}[Q_{(\lambda,\ell')}]\text{-Mod})$ is the same as the second part of the proof of Theorem 3.35, except that we now use equations (3.21) and (3.22) of Lemma 3.30 to understand the action of the braiding. $\square$

**Remark 3.37** We could define the functors $\mathcal{L}^v_{(\lambda,1)}(\Gamma)$, $\mathcal{L}^v_{(\lambda,2)}(\Gamma)$, $\mathcal{L}^v_{(\lambda,1)}(\mathcal{N})$ and $\mathcal{L}^v_{(\lambda,2)}(\mathcal{N})$ on the categories $\langle \mathcal{M}_2^+, \mathcal{M}_2^+ \rangle$ and $\langle \mathcal{M}_2^-, \mathcal{M}_2^- \rangle$ respectively (i.e. without the opposite convention for the braiding of $\mathcal{M}_2$ induced by the $^\dagger$ endofunctor; see §1.3.3). However, in this setting it is not clear that there are isomorphisms analogous to (3.26) and (3.27).

# 4. Polynomiality

In this section, we recollect the theory of polynomial functors in §4.1, then prove in §4.2–§4.3 the polynomiality results of Corollary B and Theorem C, and finally prove Corollary F in §4.4.

## 4.1. Notions of polynomiality

We review here the notions and basic properties of strong, very strong, split and weak polynomial functors. The first definitions of polynomial functors date back to Eilenberg and Mac Lane in [EM54] for functors on module categories. This notion has progressively been extended to deal
with a more general framework, and has been the object of intensive study because of its applica-
tions in representation theory (see Djament, Touzé and Vespa [DTV19]), group cohomology (see
Franjou, Friedlander, Scorichenko and Suslin [FFSS99]) and homological stability with twisted co-
efficients (see Randal-Williams and Wahl [RW17]). In particular, Djament and Vespa [DV19, §1]
introduce the notions of *strong* and *weak* polynomial functors in the context of a functor category
**Fct**(*C*, *A*), where *C* is a symmetric monoidal category where the unit is an initial object and *A* is
a Grothendieck category (see the definition below). They are then extended to the case where *C* is
pre-braided monoidal in [Sou19; Sou22], which also introduce the notion of *very strong* polynomial
functor. See also [Pal17] for a comparison of the various instances of polynomial functors. All these
notions extend verbatim to the present slightly larger framework from the previous literature on
this topic (see [Sou22, §4] for instance), the various proofs being mutatis mutandis generalisations
of these previous works. We also define the notion of *split* polynomial functor, a particular kind
of very strong polynomial functor, following an analogous notion from [RW17].

For the remainder of §4.1, we fix a strict left-module groupoid $(\mathcal{M}, \natural)$ over a braided strict monoidal groupoid $(\mathcal{G}, \natural, 0)$ satisfying the same assumptions as in §3.1.1: $(\mathcal{G}, \natural, 0)$ has no zero divisors, $\mathrm{Aut}_{\mathcal{G}}(0) = \{\mathrm{id}_0\}$, $\mathcal{M}$ and $\mathcal{G}$ are both small and skeletal, have the same set of objects identified with the non-negative integers $\mathbb{N}$ with the standard notation $\mathfrak{n}$ to denote an object, and both the monoidal and module structures $\natural$ are given on objects by addition. For example, one quickly checks that all of the examples of $\mathcal{M}$ and $\mathcal{G}$ defined in §1.3 satisfy all of these assumptions. We also consider a Grothendieck category $\mathcal{A}$, i.e. a cocomplete abelian category, which admits a generator, and in which filtered colimits of exact sequences are exact. In particular, we recall from [Gro57, §1.7, d)] that the functor category $\mathrm{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is a Grothendieck category.

**Strong, very strong and split polynomial functors.** The category of *strong* polynomial functors of degree at most $d \in \mathbb{N}$, denoted by $\mathcal{P}ol_d^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, is the full subcategory of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ defined by $\mathcal{P}ol_d^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A}) = \{0\}$ if $d < 0$ and the objects of $\mathcal{P}ol_{d-1}^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ for $d \in \mathbb{N}$ are the functors $F$ such that the functor $\delta_1(F)$ is an object of $\mathcal{P}ol_{d-1}^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. The smallest integer $d \in \mathbb{N}$ for which an object $F$ of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is an object of $\mathcal{P}ol_d^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is called the strong degree of $F$, and is denoted by $\text{sdeg}(F)$.

The category of *very strong* polynomial functors of degree at most $d \in \mathbb{N}$, denoted by $\mathcal{VPol}_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, is the full subcategory of $\mathcal{P}ol_d^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ of the objects $F$ such that $\kappa_1(F) = 0$ and the functor $\delta_1(F)$ is an object of $\mathcal{VPol}_{d-1}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$.

The category of split polynomial functors of degree at most $d \in \mathbb{N}$, denoted by $\mathcal{SPol}_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, is the full subcategory of $\mathcal{P}ol_d^{\text{str}}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ of the objects $F$ such that the translation map $i_1F: F \to \tau_1F$ is split injective in $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$.

**Remark 4.1** It follows from the definitions that split polynomiality implies very strong polyno-
miality, while very strong polynomiality implies strong polynomiality. Also, for $F$ an object of
**Fct**(*⟨G*, *M⟩*, *A*), if $F$ is very strong (resp. split) polynomial, then its strong degree $\text{sdeg}(F)$ is the
smallest integer $d \in \mathbb{N}$ such that $F$ is an object of $\mathcal{VPol}_d(\langle G, M \rangle, A)$ (resp. of $\mathcal{SPol}_d(\langle G, M \rangle, A)$).

A valuable application of these notions of polynomiality is that of *homological stability with twisted coefficients* for families of groups; see [RW17] for a detailed introduction to this notion. Namely, let us consider any pair of groupoids $\mathcal{M}$ and $\mathcal{G}$ introduced in §1.1.2.2 and §1.1.2.3, such that the family of automorphism groups of the Quillen bracket construction $\langle\mathcal{G}, \mathcal{M}\rangle$ is $\{\mathbf{B}_n(S)\}_{n\in\mathbb{N}}$ for a fixed $S \in \{\mathbb{D}, \Sigma_{g,1}, \mathbb{N}_{h,1}\}$, or $\{\mathbf{\Gamma}_{n,1}\}_{n\in\mathbb{N}}$, or $\{\mathcal{N}_{n,1}\}_{n\in\mathbb{N}}$.

**Theorem 4.2** ([RW17, Theorems D, I, 5.23, 5.26, 5.29]) *Twisted homological stability holds for the family of automorphism groups of (*⟨G*, *M⟩*) with coefficients in any object *F* of *Fct*(⟨*G*, *M⟩*, *Z*-Mod) that is strong polynomial (of finite degree *d* ∈ *N*), and for which there exists *N* ∈ *Obj(*$\mathcal{M}$*) such that, for all *r* ∈ {0, ..., *d*}, $\kappa_1(\delta_1^r F)(\mathfrak{n}) = 0$ for each $\mathfrak{n} \ge N - r$; see [RW17, Def. 4.10] for further details on these conditions. For instance, these conditions hold whenever *F* is very strong polynomial or split polynomial.*

**Remark 4.3** In the case of mapping class groups of orientable surfaces, similar twisted homological stability results to those of Theorem 4.2 from [RW17] were proven earlier by Ivanov [Iva93] and Boldsen [Bol12].
On another note, the target category of a homological representation functor $\mathcal{L}^*_{(\lambda, \ell)}$ is gener-
ally of the form $\mathbb{Z}[Q^*_{(\lambda, \ell)}]$-Mod• (see Convention 3.4). To rigorously apply Theorem 4.2 (see
Corollaries 4.9 and 4.16), we implicitly postcompose the homological representation functors by
the forgetful functor $\mathbb{Z}[Q^*_{(\lambda, \ell)}]$-Mod• $\to \mathbb{Z}$-Mod; such operations do not affect the polynomiality
properties of $\mathcal{L}^*_{(\lambda, \ell)}$.

**Weak polynomial functors.** Let $F$ be an object of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. For all objects $\mathbf{n}$ and $\mathbf{m}$ of $\mathcal{M}$, it follows from a clear diagram chase, the universal property of a kernel and the 4-lemma that the morphism $[\mathbf{m}, \mathrm{id}_{\mathbb{m}\otimes\mathbf{n}}]$ induces a canonical inclusion $\kappa_{\mathbf{n}}(F) \hookrightarrow \kappa_{\mathbb{m}\otimes\mathbf{n}}(F)$. These inclusions provide an ascending filtration on the evanescence functors $\{\kappa_{\mathbf{n}}\}_{\mathbf{n}\in\mathrm{Obj}(\mathcal{M})}$. We denote by $\kappa(F)$ the filtered colimit $\sum_{\mathbf{n}\in\mathrm{Obj}(\mathcal{M})} \kappa_{\mathbf{n}}(F)$, which is a subfunctor of $F$.

Let $K(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ be the full subcategory of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ of all those objects $F$ such that $\kappa(F) = F$. The category $K(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is a thick subcategory of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ and it is closed under colimits; see [Sou22, Prop. 4.6].

**Remark 4.4** It would be enough to assume that $\mathcal{A}$ is abelian to define strong, very strong and split polynomiality. However, we need $\mathcal{A}$ to be Grothendieck in order to work with the notion of weak polynomiality and the associated quotient category of the functor category $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. Indeed, it is necessary to assume that the filtered colimits in the category $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ are exact in order to prove [Sou22, Prop. 4.6]. See also [Sou22, Rem. 4.7] for further explanations.

Therefore, the subcategory $K(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ of the Grothendieck category $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is
localising, and we may define the associated quotient category denoted by $\mathrm{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, along
with the associated left adjoint quotient functor $\pi_{(\mathcal{G}, \mathcal{M})}$, which is exact, essentially surjective and
commutes with all colimits; see [Gab62, Chapitre III, §1].

For each object $\mathbf{n}$ of $\mathcal{M}$, the translation functor $\tau_{\mathbf{n}}$ and the difference functor $\delta_{\mathbf{n}}$ in the category $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ induce exact endofunctors of $\mathrm{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, which commute with colimits, respectively called again the translation functor $\tau_{\mathbf{n}}$ and the difference functor $\delta_{\mathbf{n}}$. In addition, we have the commutation relations $\delta_{\mathbf{n}} \circ \pi_{(\mathcal{G}, \mathcal{M})} = \pi_{(\mathcal{G}, \mathcal{M})} \circ \delta_{\mathbf{n}}$ and $\tau_{\mathbf{n}} \circ \pi_{(\mathcal{G}, \mathcal{M})} = \pi_{(\mathcal{G}, \mathcal{M})} \circ \tau_{\mathbf{n}}$. Therefore, the exact sequence (3.1) induces a short exact sequence $\mathrm{Id} \hookrightarrow \tau_{\mathbf{n}} \to \delta_{\mathbf{n}}$ for the induced endofunctors of $\mathrm{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. Finally, the endofunctors $\delta_1, \delta_n, \tau_1$ and $\tau_n$ of $\mathrm{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ pairwise commute up to natural isomorphism coming from the braiding.

We then define, inductively on $d \in \mathbb{N}$, the category of polynomial functors of degree at most $d$, denoted by $\mathcal{P}ol_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, to be the full subcategory of $\mathbf{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ as follows. If $d < 0$, $\mathcal{P}ol_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A}) = \{0\}$; if $d \ge 0$, the objects of $\mathcal{P}ol_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ are the functors $F$ such that the functor $\delta_1(F)$ is an object of $\mathcal{P}ol_{d-1}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. For an object $F$ of $\mathbf{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ that is polynomial of degree at most $d \in \mathbb{N}$, the smallest integer $n \le d$ for which $F$ is an object of $\mathcal{P}ol_n(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is called the degree of $F$. An object $F$ of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$ is weak polynomial of degree at most $d$ if its image $\pi_{(\mathcal{G}, \mathcal{M})}(F)$ is an object of $\mathcal{P}ol_d(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. The degree of polynomiality of $\pi_{(\mathcal{G}, \mathcal{M})}(F)$ is called the weak degree of $F$, and is denoted by $\mathrm{wdeg}(F)$.

**Lemma 4.5** Let $F$ be an object of $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$. If $F$ is strong polynomial of strong degree $\mathrm{sdeg}(F)$, then it is weak polynomial of weak degree $\mathrm{wdeg}(F) \le \mathrm{sdeg}(F)$.

*Proof.* The result is a straightforward consequence of the commutation relation $\delta_1^p(\pi_{(\mathcal{G}, \mathcal{M})}(F)) = \pi_{(\mathcal{G}, \mathcal{M})}(\delta_1^p(F))$ for all $p \ge 1$. $\square$

**Remark 4.6** Since each object $\mathbf{n}$ is equal to $1^{2n}$ in the category $\langle\mathcal{G}, \mathcal{M}\rangle$, our definitions of the notions of strong, very strong, split and weak polynomiality are equivalent to the more classical ones with the analogous criteria on the functors $\tau_{\mathbf{n}}$, $\delta_{\mathbf{n}}$ and $\kappa_{\mathbf{n}}$ for all $\mathbf{n} \in \mathrm{Obj}(\mathcal{G})$ (instead of just $\mathbf{n} = 1$); see [DV19, Prop. 1.8], [Sou19, Prop. 3.9] and [Sou22, Prop. 4.4, (2)] for further details.

Furthermore, for a short exact sequence $0 \to F \to G \to H \to 0$ in the category $\mathbf{Fct}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, the snake lemma induces the following short exact sequence in $\mathbf{St}(\langle\mathcal{G}, \mathcal{M}\rangle, \mathcal{A})$, which is the image of (3.2):

$$
0 \to \delta_n \circ \pi_{(\mathcal{G}, \mathcal{M})}(F) \to \delta_n \circ \pi_{(\mathcal{G}, \mathcal{M})}(G) \to \delta_n \circ \pi_{(\mathcal{G}, \mathcal{M})}(H) \to 0. \quad (4.1)
$$

Finally, we recall useful properties of the categories associated with the different types of polynomial
functors, which are proven in [Sou22, Props. 4.4, 4.10] (split polynomial functors are not considered
there, but their study follows repeating mutatis mutandis this reference).
**Proposition 4.7** Let $d \ge 0$ be an integer. The categories $\mathcal{P}ol_d^{str}((\mathcal{G}, \mathcal{M}), \mathcal{A})$, $\mathcal{VPol}_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ and $\mathcal{SPol}_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ are closed under the translation functor, direct sum and direct summand. The categories $\mathcal{P}ol_d^{str}((\mathcal{G}, \mathcal{M}), \mathcal{A})$ and $\mathcal{VPol}_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ are closed under extensions. The category $\mathcal{P}ol_d^{str}((\mathcal{G}, \mathcal{M}), \mathcal{A})$ is closed under colimits. The categories $\mathcal{VPol}_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ and $\mathcal{SPol}_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ are closed under normal subobjects (i.e. kernels of epimorphisms). As a subcategory of $\mathbf{St}((\mathcal{G}, \mathcal{M}), \mathcal{A})$, the category $\mathcal{P}ol_d((\mathcal{G}, \mathcal{M}), \mathcal{A})$ is thick, complete and cocomplete.

## 4.2. For surface braid group functors

In this section, we prove the polynomiality properties of Corollary B and Theorem C for the homological representation functors for classical braid groups and surface braid groups defined in §1.3.1 and §1.3.2. Throughout §4.2, we consider homological representation functors indexed by an ordered partition $\lambda \vdash k$ of an integer $k \ge 1$ and by the depth $\ell \ge 1$ of a lower central series.

### 4.2.1. Classical homological representation functors

We prove here Corollary B in the classical (i.e. non-vertical) setting for homological representation functors. These polynomiality results actually hold both for the standard functors (i.e. $\mathfrak{L}\mathfrak{B}_{(\lambda, \ell)}$ and $\mathfrak{L}_{(\lambda, \ell)}(S)$) as well as for their *untwisted* versions (i.e. $\mathfrak{L}\mathfrak{B}^u_{(\lambda, \ell)}$ and $\mathfrak{L}^u_{(\lambda, \ell)}(S)$).

*Proof of Corollary B for classical braid groups.* Following §3.2.2, we consider the functor $\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ where $*$ either stands for the blank space or $*$ = $u$.

We first consider the case of $k=1$, and so $\lambda = (1)$. We recall that the preliminary study of §3.2.1 (except (3.6) and (3.7)) holds for the functor $\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$. Following Notation 3.12, for each $n \in \text{Obj}(\beta)$, the $\mathbf{B}_n$-representation $\delta_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}(n)$ is the free $\mathbb{Z}[Q^*_{((1),\ell)}(\mathbb{D})]$-module of rank one, with generators given by the tuples $(w_0, w_1, \dots, w_{n-1})$ such that $|w_0| = 1$, while $\kappa_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)} = 0$. By expressing $\delta_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}([1, \text{id}_{1\mathbb{Z}_n}])$ as a quotient map of $\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}(\sigma_1^{-1}) \circ \mathfrak{L}\mathfrak{B}^*_{((1),\ell)}([1, \text{id}_{1\mathbb{Z}_n}])$, it follows from Lemma 3.16 and from the description (3.5) of $\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}([1, \text{id}_{1\mathbb{Z}_n}])$ that $i_1(\delta_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)})(n)$ is an isomorphism for $n \ge 1$, while $i_1(\delta_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)})(0)$ is the zero map. So $\kappa_1\delta_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)} = 0$, while $\delta_1^2\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$ is the atomic functor whose unique non-null value is $\delta_1^2\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}(0) = \mathbb{Z}[Q^*_{((1),\ell)}(\mathbb{D})]$. A fortiori, $\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$ is strong polynomial of strong degree 2 and weak polynomial of weak degree 1. Furthermore, it follows from the commutation property of $\delta_1$ with $\tau_1$ that $\delta_1(\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)})(n) = \mathbb{Z}[Q^*_{((1),\ell)}(\mathbb{D})]$ for each $n \ge 0$. By repeating mutatis mutandis the above argument, we show that $\kappa_1(\delta_1(\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)})) = \delta_1^2(\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}) = 0$, and thus $\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$ is both very strong and weak polynomial, of both strong and weak degrees 1. Moreover, by applying Lemma 3.9 and then repeating verbatim the above argument, we deduce that this polynomiality result for $\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$ still holds after any non-zero change of rings operation.

Now, we proceed by induction on $k \ge 2$, reasoning on each ordered partition $\lambda \vdash k$. First, we consider the case of $k=2$ (with an ordered partition $\lambda \vdash 2$). It follows from Theorem 3.17 that $\kappa_1(\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}) = 0$, while the difference functor $\delta_1\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ is determined by $\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$ (potentially up to a non-zero change of rings; see Convention 3.8). By the above polynomiality results on $\tau_1\mathfrak{L}\mathfrak{B}^*_{((1),\ell)}$, we deduce that the functor $\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ is both very strong and weak polynomial, of both strong and weak degrees 2. By using Corollary 3.10 and repeating verbatim this argument, this polynomiality result for $\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ with $\lambda \vdash 2$ also holds after any non-zero change of rings operation.

We do the inductive step on any fixed $\lambda \vdash k \ge 2$. Namely, we assume that for each $\lambda' \in \{\lambda-1\}$ (see Notation 3.2), the functor $\mathfrak{L}\mathfrak{B}^*_{(\lambda', \ell)}$ is both very strong and weak polynomial, of both strong and weak degrees $k-1$, and that these properties still hold after any non-zero change of rings operation. Since $\mathcal{VPol}_d((\beta, \beta), \mathbb{Z}[Q^*_{(\lambda, \ell)}(\mathbb{D})]$-Mod$^*$) is closed under $\tau_1$ and under normal subobjects by Proposition 4.7, this inductive assumption implies that $\tau_1\mathfrak{L}\mathfrak{B}^*_{(\lambda', \ell)}$ is both very strong and weak polynomial, of both strong and weak degrees $k-1$. Now, we deduce from Theorem 3.17 that $\kappa_1\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)} = 0$, while the difference functor $\delta_1\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ is a direct sum of functors of the form $\tau_1\mathfrak{L}\mathfrak{B}^*_{(\lambda', \ell)}$ where $\lambda' \in \{\lambda-1\}$ (potentially up to a non-zero change of rings; see Convention 3.8). Therefore, the functor $\mathfrak{L}\mathfrak{B}^*_{(\lambda, \ell)}$ is both very strong and weak polynomial, of both strong and weak degrees $k$. This polynomiality result also holds after applying any non-zero change of rings operation by using Corollary 3.10 and repeating verbatim the above reasoning, which ends the induction. $\square$
**Remark 4.8** The functors $\mathfrak{L}\mathfrak{B}_{((1),2)} \otimes \mathbb{C}[\mathbb{Z}]$ and $\mathfrak{L}\mathfrak{B}_{((2),2)} \otimes \mathbb{C}[\mathbb{Z}^2]$ correspond to the reduced Burau functor $\mathfrak{B}$ and the Lawrence-Krammer functor $\mathfrak{L}_k$ defined in [Sou19, §1.2]. By Corollary 3.10, the polynomiality results of Corollary B recover those of [Sou19, Props. 3.25 and 3.33] for the functors $\mathfrak{L}\mathfrak{B}_{((1),2)} \otimes \mathbb{C}[\mathbb{Z}]$ and $\mathfrak{L}\mathfrak{B}_{((2),2)} \otimes \mathbb{C}[\mathbb{Z}^2]$. Also, the corresponding key short exact sequences using [Sou19, §1.2] (i.e. (3.13) for $\lambda \in \{1,2\}$ and $\ell = 2$) are proven via an alternative method, which is more algebraic than that of Theorem 3.17.

*Proof of Corollary B for surface braid groups.* Following §3.2.3, we use here the notation $\mathcal{L}^*_{(\lambda,\ell)}(S)$ where $\star$ either stands for the blank space or $\star = u$ and $S$ is either $\Sigma_{g,1}$ or $\mathcal{N}_{h,1}$ with $g, h \ge 1$.

First, we consider the case of $k=1$ (with the trivial partition $\lambda = (1)$). We recall that the preliminary study of §3.2.1 (except (3.6) and (3.7)) holds for the functor $\mathcal{L}^*_{((1),\ell)}(S)$. Following Notation 3.12, for each $n \in \text{Obj}(\beta^S)$, the $\mathbb{Z}[Q^*_{((1),\ell)}(S)]$-module $\mathcal{L}^*_{((1),\ell)}(S)(\mathfrak{n})$ is free of rank $g_S+n-1$, with generators denoted by tuples $(w_1, \dots, w_{g_S+n-1})$, where the $w_i \in \{0,1\}$ are integers such that $\sum_{1 \le i \le g_S+n-1} w_i = 1$. Also, the map $i_1(\mathcal{L}^*_{((1),\ell)}(S))_n = \mathcal{L}^*_{((1),\ell)}(S)([1, \text{id}_{1\mathbb{Z}_n}])$ is the injection defined by $(w_1, \dots, w_{g_S+n-1}) \mapsto (0, w_1, \dots, w_{g_S+n-1})$. Hence the cokernel $\delta_1 \mathcal{L}^*_{((1),\ell)}(S)(\mathfrak{n})$ is the free $\mathbb{Z}[Q^*_{((1),\ell)}(S)]$-module of rank one generated by the element $(1, 0, \dots, 0)$ if $n \ge 1$, while $\delta_1 \mathcal{L}^*_{((1),\ell)}(S)(\mathfrak{0})$ is a free $\mathbb{Z}[Q^*_{((1),\ell)}(S)]$-module of rank $g_S$. By expressing $\delta_1 \mathcal{L}^*_{((1),\ell)}(S)([1, \text{id}_{1\mathbb{Z}_n}])$ as a quotient map of $\mathcal{L}^*_{((1),\ell)}(S)(\sigma_1^{-1}) \circ \mathcal{L}^*_{((1),\ell)}(S)([1, \text{id}_{1\mathbb{Z}_n}])$, it follows from Lemma 3.16 that $i_1(\delta_1 \mathcal{L}^*_{((1),\ell)}(S))_n$ is an isomorphism for $n \ge 1$, and a fortiori $\delta_1^2(\mathcal{L}^*_{((1),\ell)}(S))(\mathfrak{n}) = 0$ when $n \ge 1$. Therefore, the functor $\delta_1^2(\mathcal{L}^*_{((1),\ell)}(S))$ is either null or atomic (with unique non-null value being the image of $\mathfrak{0}$), and so $\mathcal{L}^*_{((1),\ell)}(S)$ is weak polynomial of weak degree 1. Moreover, the target of the map $i_1(\delta_1^2 \mathcal{L}^*_{((1),\ell)}(S))_n$ is 0 for all $n \ge 0$, so $\delta_1^3 \mathcal{L}^*_{((1),\ell)}(S) = 0$ and thus $\mathcal{L}^*_{((1),\ell)}(S)$ is strong polynomial, of strong degree at most 2.

Now, we prove by induction on $k \ge 1$ that, for each ordered partition $\lambda \vdash k$, the functor $L_\lambda := (\tau_1 \mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2}$ is weak polynomial with wdeg$(L_\lambda) = k$ and strong polynomial with sdeg$(L_\lambda)$ equal to $k$ or $k+1$, and that these properties still hold after any non-zero change of rings operation. The base case corresponds to studying the polynomiality of $L_{(1)} := (\tau_1 \mathcal{L}^*_{((1),\ell)}(S))_{|\ge 2}$. Repeating mutatis mutandis the first arguments above, we deduce that $\kappa_1 L_{(1)} = 0$ and $\delta_1(L_{(1)})(\mathfrak{n})$ is the free $\mathbb{Z}[Q^*_{((1),\ell)}(S)]$-module of rank one generated by the element $(1, 0, \dots, 0)$ if $n \ge 1$, while $\delta_1(L_{(1)})(\mathfrak{0})$ is the free $\mathbb{Z}[Q^*_{((1),\ell)}(S)]$-module of rank $g_S+1$ on the generators $(w_1, \dots, w_{g_S+1})$ with $w_i \in \{0,1\}$. Viewing $\delta_1 L_{(1)}([1, \text{id}_{1\mathbb{Z}_n}])$ as a quotient map of $L_{(1)}(\sigma_1^{-1}) \circ L_{(1)}([1, \text{id}_{1\mathbb{Z}_n}])$, we deduce from Lemma 3.16 that $i_1(\delta_1 L_{(1)})(\mathfrak{n})$ is an isomorphism for $n \ge 1$. Hence we have $\delta_1^2(L_{(1)})(\mathfrak{n}) = 0$ when $n \ge 1$, and the target of the map $i_1(\delta_1^2 L_{(1)})$ is 0 for all $n \ge 0$. Therefore, as above, the functor $L_{(1)}$ is weak polynomial with wdeg$(L_{(1)}) = 1$, and strong polynomial with sdeg$(L_{(1)})$ equal to 1 or 2 by Lemma 4.5. Furthermore, since $\delta_1 L_{(1)}(\mathfrak{n})$ is a free $\mathbb{Z}[Q^*_{(\lambda,\ell)}(D)]$-module for each $\mathfrak{n} \in \text{Obj}(\beta^S)$ and $\kappa_1 L_{(1)} = 0$, we deduce that this polynomiality result for $L_{(1)}$ still holds after any non-zero change of rings operation by using Lemma 3.9 and then repeating verbatim the above argument.

We do the inductive step on a fixed ordered partition $\lambda \vdash k$ with $k \ge 1$. Namely, we assume that for each $\lambda' \in \{\lambda-1\}$ (see Notation 3.2), the functor $L_\lambda := (\tau_1 \mathcal{L}^*_{(\lambda',\ell)}(S))_{|\ge 2}$ is weak polynomial of weak degree $k-1$ and strong polynomial of strong degree $k-1$ or $k$, and that these properties still hold after any non-zero change of rings operation. By the inductive assumption on $L_{\lambda'}$, the commutation properties of $\delta_1$ with colimits and with $\pi_{(\beta,\beta^S)}$ and the right-exactness of $\pi_{(\beta,\beta^S)}$, we deduce that the functor $\bigoplus_{1 \le j \le r} L_{\lambda[j]}$ (as well as its versions after applying any non-zero change of rings operations to each one of the $L_{\lambda[j]}$ factors) is weak polynomial of weak degree $k-1$ and strong polynomial of strong degree $k-1$ or $k$. By Theorem 3.24, the difference functor $\delta_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})$ is an extension of the atomic functor $\tau_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})(1)$ by $\bigoplus_{1 \le j \le r} L_{\lambda[j]}$ (potentially up to a non-zero change of rings; see Convention 3.8). It thus follows from the properties on extensions of Proposition 4.7 that the functor $\delta_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})$ is weak polynomial of weak degree at most $k-1$, and strong polynomial of strong degree at most $k$. In the stable category $\text{St}(\langle\beta, \beta^S\rangle, \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-Mod$^\bullet$), we note that $\pi_{(\beta,\beta^S)}(\tau_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})(1)) = 0$ because $\kappa(\tau_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})(1)) = \tau_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})(1)$. Hence $\pi_{(\beta,\beta^S)}(\delta_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})) \cong \bigoplus_{1 \le j \le r} \pi_{(\beta,\beta^S)}(L_{\lambda[j]})$ and thus the weak degree of $\delta_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})$ is equal to $k-1$. Also, by Lemma 4.5, the strong degree of $\delta_1(\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})$ is thus $k-1$ or $k$. Therefore, the functor $\mathcal{L}^*_{(\lambda,\ell)}(S))_{|\ge 2})$ is weak polynomial of weak degree $k$, and strong polynomial
of strong degree $k$ or $k+1$. Then, by the properties on extensions of Proposition 4.7, we deduce from Corollary 3.25 that the functor $L_\lambda$ is weak polynomial of weak degree at most $k$, and strong polynomial of strong degree at most $k+1$. Since $\delta_1$ commutes with colimits, the inductive assumption on the functors $L_{\lambda[j]}$ implies that $\delta_1^k \pi_{(\beta, \beta^S)} (\bigoplus_{1 \le j \le r} L_{\lambda[j]}) = 0$. Then, by iterating the short exact sequence (4.1) on (3.18), we deduce that the functor $\delta_1^k \pi_{(\beta, \beta^S)} (L_\lambda)$ is isomorphic to $\delta_1^k \pi_{(\beta, \beta^S)} (\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2})$, which is non-null since $\mathrm{wdeg}(\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2}) = k$. Hence $\mathrm{wdeg}(L_\lambda) = k$, and thus $\mathrm{sdeg}(L_\lambda)$ is equal to $k$ or $k+1$ by Lemma 4.5. These polynomiality results for $\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2}$ and $L_\lambda$ also hold after applying any non-zero change of rings operation, by using Corollary 3.10 and repeating verbatim the above reasoning, which ends the induction.

In particular, we prove in the process of the induction that, for each $k \ge 1$ and each ordered partition $\lambda \vdash k$, the functor $\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2}$ is weak polynomial of weak degree $k$, and strong polynomial of strong degree $k$ or $k+1$. Now, recall from §3.2.3 that $\mathcal{L}_{(\lambda, \ell)}^*(S)$ is an extension of the atomic functor $\mathcal{L}_{(\lambda, \ell)}^*(S)(1)$ by $\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2}$, i.e there is a short exact sequence $\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2} \hookrightarrow \mathcal{L}_{(\lambda, \ell)}^*(S) \to \mathcal{L}_{(\lambda, \ell)}^*(S)(1)$. By the properties on extensions of Proposition 4.7, the functor $\mathcal{L}_{(\lambda, \ell)}^*(S)$ is weak polynomial of weak degree with $\mathrm{wdeg}(\mathcal{L}_{(\lambda, \ell)}^*(S)) \le k$, and strong polynomial with $\mathrm{sdeg}(\mathcal{L}_{(\lambda, \ell)}^*(S)) \le k+1$. Then we have $\pi_{(\beta, \beta^S)}(\mathcal{L}_{(\lambda, \ell)}^*(S)(1)) = 0$ because $\kappa(\mathcal{L}_{(\lambda, \ell)}^*(S)(1)) = \mathcal{L}_{(\lambda, \ell)}^*(S)(1)$, so $\pi_{(\beta, \beta^S)}(\mathcal{L}_{(\lambda, \ell)}^*(S)) \cong \pi_{(\beta, \beta^S)}(\mathcal{L}_{(\lambda, \ell)}^*(S)|_{\ge 2})$. We thus deduce that $\mathrm{wdeg}(\mathcal{L}_{(\lambda, \ell)}^*(S)) = k$, and so $\mathrm{sdeg}(\mathcal{L}_{(\lambda, \ell)}^*(S)) = k$ or $k+1$ by Lemma 4.5. $\square$

**Corollary 4.9** *Twisted homological stability holds for the classical braid groups and surface braid groups with coefficients in the homological representation functors of Theorems 3.17 and 3.24.*

*Proof.* Following §3.2.1, we use the generic notation $\mathcal{L}_{(\lambda, \ell)}^*(S)$ to study the functors $\mathcal{L}\mathfrak{B}^*(\lambda, \ell)$ and $\mathcal{L}^*(\lambda, \ell)(S)$ of Theorems 3.17 and 3.24. Recall from the description (3.5) of the map $i_1(\mathcal{L}^*(\lambda, \ell)(S))_n$ that $\kappa_1 \mathcal{L}^*(\lambda, \ell)(S)(n) = 0$ for all $n \ge 2$. Then, using the commutation property of $\delta_1$ with $\tau_1$ (and noting that $\delta_1(\mathcal{L}^*(\lambda, \ell)(S)|_{\ge 2})(n) \cong \delta_1(\mathcal{L}^*(\lambda, \ell)(S))(n)$ by (3.2) for $n \ge 2$ when $S \ne \mathbb{D}$), it follows from a clear iteration of the short exact sequences (3.13) and (3.17) that $\kappa_1(\delta_1^r \mathcal{L}^*(\lambda, \ell)(S))(n) = 0$ for all $n \ge 2$ and all $r \ge 0$. By this last property along with Corollary B, each functor $\mathcal{L}^*(\lambda, \ell)(S)$ thus satisfies the condition of Theorem 4.2 (as long as we choose $N \ge 2+d$), whence the result. $\square$

**Remark 4.10** In the above proof of Corollary B for surface braid groups, we have not determined if the strong degree of $\mathcal{L}_{(\lambda, \ell)}^*(S)$ is $k$ or $k+1$, nor addressed the question of whether or not this functor is *very* strong polynomial. This is actually an aftereffect of the difficulty of computing the first entries $\delta_1^m(\mathcal{L}_{(\lambda, \ell)}^*(S))(0)$ and $\kappa_1(\delta_1^{m-1}(\mathcal{L}_{(\lambda, \ell)}^*(S))(0)$ for $m \ge 2$ with the techniques of the present paper. Indeed, although it is not difficult to compute $\delta_1^2(\mathcal{L}_{(\lambda, \ell)}^*(S))(n)$ and check that $\kappa_1(\delta_1 \mathcal{L}_{(\lambda, \ell)}^*(S))(n) = 0$ for $n \ge 1$ via the methods of §3.2.1 and §3.2.3, the map

$$i_1(\delta_1 \mathcal{L}_{(\lambda, \ell)}^*(S))_0 : (\delta_1 \mathcal{L}_{(\lambda, \ell)}^*(S))(0) \cong \mathcal{L}_{(\lambda, \ell)}^*(S)(1) \to (\delta_1 \mathcal{L}_{(\lambda, \ell)}^*(S))(1)$$

is however much trickier to study. Let us illustrate this with the case of $k=1$, $\lambda=(1)$, $\ell=2$ and $S = \Sigma_{1,1}$, for which $Q_{((1),2)}(S) \cong \mathbb{Z}^2 \cong \langle A \rangle \oplus \langle B \rangle$ (see Remark 1.18). We recall that the generators $\{[1,0], [0,1]\}$ form a basis of $\mathcal{L}_{((1),2)}(\Sigma_{1,1})(1) \cong \mathbb{Z}[\mathbb{Z}^2]^\oplus 2$. Using the further techniques of [PS], one may then prove that the kernel of $i_1(\delta_1 \mathcal{L}_{((1),2)}(S))_0: \mathbb{Z}[\mathbb{Z}^2]^\oplus 2 \to \mathbb{Z}[\mathbb{Z}^2]$ is isomorphic to the free $\mathbb{Z}[\mathbb{Z}^2]$-submodule of $\mathbb{Z}[\mathbb{Z}^2]^\oplus 2$ generated by $(1-A)[0,1] - (1-B)[1,0]$, while $\delta_1^2(\mathcal{L}_{((1),2)}(S))(0) \cong \mathbb{Z}[\mathbb{Z}^2]/(1-A, 1-B)$. This along with the above proof of Corollary B proves that $\mathcal{L}_{((1),2)}(\Sigma_{1,1})$ is strong polynomial of degree 2 but not *very* strong polynomial. Nevertheless, note that if we apply the change of rings functor for the homomorphism $q: \mathbb{Z}[\mathbb{Z}^2] \to \mathbb{Q}(\mathbb{Z}^2)$ (where $\mathbb{Q}(\mathbb{Z}^2)$ is the field of fractions of $\mathbb{Z}[\mathbb{Z}^2]$), we deduce that the functor $q_1\mathcal{L}_{((1),2)}(\Sigma_{1,1})$ is strong polynomial of degree 1. The strong degree thus decisively depends on the ground ring of the functor in this case. Furthermore, analogous results may be proved for more general $\mathcal{L}_{(\lambda, \ell)}^*(S)$.

In contrast, by similar work to §3.2.3 and the proof of Corollary B, it is routine to check that the shifted-by-1 functor $\tau_1 \mathcal{L}_{(\lambda, \ell)}^*(S)$ is very strong and weak polynomial of both strong and weak degrees $k$. This exemplifies how the strong polynomial degree may be heavily affected by the low values of a given functor, thus not being optimal to describe its global behaviour, in particular for homological stability. Moreover, this shows the interest of the notion of weak polynomiality since it reflects more accurately the stable behaviour of functors.
### 4.2.2. Vertical-type alternatives

We now deal with the vertical-type alternatives of the homological representation functors of the surface braid groups. Following the framework of §3.2, we consider the generic vertical homological representation functor $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ where $\star$ either stands for the blank space or $\star = u$, $S \in \{\mathbb{D}, \Sigma_{g,1}, \mathcal{N}_{h,1}\}$ with $g \ge 1$ and $h \ge 1$ and the associated transformation group is denoted by $Q^*_{(\lambda,\ell)}(S)$. We recall from Proposition 2.9 that, for each $n \in \text{Obj}(\beta^S)$, the $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n)$ is free with basis indexed by the set of ("vertical") tuples $w^v$ as pictured in Figures 2.3a and 2.3b, which has the same dimension as the ("classical") functor $\mathcal{L}^*_{(\lambda,\ell)}(S)(n)$.

First of all, we focus on a significant fact about the behaviour of the functor $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ after applying the operation $\delta_1$.

**Lemma 4.11** The functor $\delta_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ sends every morphism that is not an endomorphism to zero.

*Proof.* Recall that, by construction (see Lemma 1.11), the morphism $n \to 1_{\mathfrak{h}}$ of the domain category $\langle\beta, \beta^S\rangle$ is sent, under each of our functors $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$, to the map on Borel-Moore homology induced by the evident inclusion of configuration spaces. Since every morphism of the domain category that is not an endomorphism factors through one of these canonical morphisms, it suffices to show that all of these are sent to zero under $\delta_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$. In other words, we wish to show that the map labelled by $(*)$ in the following diagram is zero, where the rows are exact:

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n) \arrow[r] \arrow[d] & \tau_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n) \arrow[r] \arrow[d] & \delta_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n) \arrow[r] \arrow[d] & 0 \\
\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(1_{\mathfrak{h}}) \arrow[r, "(\dagger)"'] \arrow[d] & \tau_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(1_{\mathfrak{h}}) \arrow[r, "(\ddagger)"'] \arrow[d] & \delta_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(1_{\mathfrak{h}}) \arrow[r, "(*)"'] \arrow[d] & 0
\end{tikzcd} \quad (4.2) $$

To do this, it suffices to show that there is a diagonal morphism making the two triangles commute. Recalling that $\tau_1 F(n) = F(1_{\mathfrak{h}})$ in general, we will be able to take the diagonal morphism to be the identity as long as the two maps labelled $(\dagger)$ and $(\ddagger)$ are equal (we note that the top-left horizontal map $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n) \to \tau_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n)$ and the left-most vertical map $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(n) \to \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)(1_{\mathfrak{h}})$ in (4.2) are always equal by definition of the natural transformation $\text{Id} \to \tau_1$).

By definition of $\tau_1 \mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$, its action on the canonical morphism $[1, \text{id}_{1_{\mathfrak{h}}}] : n \to 1_{\mathfrak{h}}$ is given by the action of $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ on the canonical morphism $[1, \text{id}_{2_{\mathfrak{h}}}] : 1_{\mathfrak{h}} \to 2_{\mathfrak{h}}$ composed with $(b_{1,1}^{\beta})^{-1}\text{id}_n$, where $b_{1,1}^{\beta}$ is the braiding $1_{\mathfrak{h}} \cong 1_{\mathfrak{h}}$ of the groupoid $\beta$; see (1.2). This describes the map $(\dagger)$; on the other hand, the map $(\ddagger)$ is given simply by the action of $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ on the canonical morphism $1_{\mathfrak{h}} \to 2_{\mathfrak{h}}$. It is therefore enough to prove that the automorphism $b_{1,1}^{\beta}\text{id}_n$, which canonically identifies with the Artin generator $\sigma_1$, acts by the identity on the image of $(\ddagger)$. This is immediate from Figure 4.1, where the image of an arbitrary basis element $w^v = (w_1, \dots, w_n)$ under $(\ddagger)$ is depicted in green (supported on the vertical arcs) and the support of a diffeomorphism representing the mapping class $\sigma_1$ is shaded in grey. Since these supports are disjoint, the action of $b_{1,1}^{\beta}\text{id}_n$ on the image of $(\ddagger)$ is trivial. $\square$

**Remark 4.12** It is instructive to consider why the same argument does not also show that the functor $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}(S)$ sends every canonical morphism $n \to 1_{\mathfrak{h}}$ to the zero morphism. This boils down to the fact that, in the analogue of Figure 4.1 for the *non-vertical* version $\mathcal{L}^*_{(\lambda,\ell)}(S)$ of the functor, the supports are not disjoint.

We are now ready to prove the (non-)polynomiality results of Theorem C, which actually hold for any one of these vertical-type alternatives $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ (i.e. also for the *untwisted* versions). In particular, this shows that these vertical-type alternatives exhibit unexpected interesting behaviour with respect to polynomiality, which thoroughly differs from their “classical” (non-vertical) counterparts studied in §3.2.2–§3.2.3. Indeed, they are not strong polynomial, which is a counter-intuitive property since the dimensions of the representations encoded by each $\mathcal{L}^{*,v}_{(\lambda,\ell)}(S)$ grow in the same polynomial way as those of $\mathcal{L}^*_{(\lambda,\ell)}(S)$.
<figure><img src="image_21.png" /><figcaption>Figure 4.1 The support of a diffeomorphism representing the mapping class $\sigma_1 = b_{1,1}^{\beta} \sharp \mathrm{id}_n$ and the image of an arbitrary basis element $\mathbf{w}^v = (w_1, \dots, w_n)$, shown in green, under the map $\S$ of (4.2).</figcaption></figure>

*Proof of Theorem C for surface braid groups.* By similar reasoning to that of §3.2.1, we see that the map $\mathcal{L}_{(\lambda,\ell)}^{*,v}(S)([1, \mathrm{id}_{\sharp n}])$ is the injection defined on basis elements by $(w_1, \dots, w_{g_S+n-1})^v \mapsto (\emptyset, w_1, \dots, w_{g_S+n-1})^v$, so it follows that $\delta_1 \mathcal{L}_{(\lambda,\ell)}^{*,v}(S)(\mathfrak{n})$ is a non-trivial free $\mathbb{Z}[Q_{(\lambda,\ell)}^*(S)]$-module with the same dimension as $\delta_1 \mathcal{L}_{(\lambda,\ell)}^*(S)(\mathfrak{n})$ for all $\mathfrak{n} \ge 1$. Meanwhile it follows from Lemma 4.11 that $\delta_1 \mathcal{L}_{(\lambda,\ell)}^{*,v}(S)$ assigns the zero map to all morphisms of $\langle \beta, \beta^S \rangle(\mathfrak{n}, \mathfrak{m})$ with $\mathfrak{n} \ne \mathfrak{m}$. So $\delta_1 \mathcal{L}_{(\lambda,\ell)}^{*,v}(S)$ is isomorphic to a direct sum of infinitely many atomic functors. It follows that $\delta_1^m \mathcal{L}_{(\lambda,\ell)}^{*,v}(S) \ne 0$ for any $m \in \mathbb{N}$ while $\pi_{\langle \beta, \beta^S \rangle}(\delta_1 \mathcal{L}_{(\lambda,\ell)}^*(S)) = 0$ in the stable category $\mathrm{St}(\langle \beta, \beta^S \rangle, \mathbb{Z}[Q_{(\lambda,\ell)}^*(S)]-\mathrm{Mod}^\bullet)$, whence the result. $\square$

**Remark 4.13** The first steps of the proofs of Theorems 3.17 and 3.24 do go through in the vertical setting, inducing a short exact sequence of functors defined on the groupoid $\beta^S$ for each $\mathcal{L}_{(\lambda,\ell)}^{*,v}(S)$, analogous to (3.13) and (3.17) but only at the level of automorphism groups.

Finally, we briefly deal with the duals of the homological representations of Theorems 3.17 and 3.24. Let us consider any one of the above homological representation functors $\mathcal{L}_{(\lambda,\ell)}^*(S)$. By Corollary 2.14, the $\mathbf{B}_n(S)$-representation $H_k^\beta(C_\lambda(\mathbb{D}_n\sharp S); \mathbb{Z}[Q_{(\lambda,\ell)}^*(S)] \otimes \mathcal{O})$ of §2.3 is the dual representation of $\mathcal{L}_{(\lambda,\ell)}^*(S)(\mathfrak{n})$. Gathering these representations and assigning for each $[\mathfrak{m}, \mathrm{id}_{\mathrm{m}\sharp\mathfrak{n}}] \in \langle \beta, \beta^S \rangle$ the evident analogue of the map $\iota_{\mathfrak{m},\mathfrak{n}}$ of §1.2.4 for homology relative to the boundary, one may easily prove the analogue of Lemma 1.11 so that we define a functor $\mathcal{L}_{(\lambda,\ell)}^{*,\vee}(S): \langle \beta, \beta^S \rangle \to \mathbb{Z}[Q_{(\lambda,\ell)}^*(S)]-\mathrm{Mod}^\bullet$. Then the reasoning of the proof of Theorem C repeats verbatim:

**Theorem 4.14** The functor $\mathcal{L}_{(\lambda,\ell)}^{*,\vee}(S)$ is not strong polynomial, but is weak polynomial of weak degree 0.

### 4.3. For mapping class group functors

In this section, we prove the polynomiality results of Corollary B and Theorem C for the homological representation functors for mapping class groups defined in §1.3.3. Following §3.3, we use the generic notation $\mathcal{L}_{(\lambda,\ell)}^*$ for any one of the functors (1.17) and (1.18) indexed by an ordered partition $\lambda \vdash k$ of an integer $k \ge 1$ and by the depth $\ell \ge 1$ of a lower central series, $\mathcal{L}_{(\lambda,\ell)}^{*,v}$ for the vertical-type alternative functor, $Q_{(\lambda,\ell)}^*(S)$ with $S \in \{\mathbb{T}, \mathbb{M}\}$ for the associated transformation group and $\mathcal{M}$ for either $\mathcal{M}_2^+$ or $\mathcal{M}_2^-$.

*Proof of Corollary B and Theorem C for mapping class groups.* We proceed by induction on $k \ge 1$, reasoning on each ordered partition $\lambda \vdash k$ and considering the functor $\mathcal{L}_{(\lambda,\ell)}^*$. First, we consider the case of $k=1$ with $\lambda = (1)$. We recall that the preliminary study of §3.3.1 up to the paragraph “Difference functor decomposition” holds for $\mathcal{L}_{((1)),\ell)}^*$. Hence $\kappa_1 \mathcal{L}_{((1),\ell)}^* = 0$, while the MCG$(S^{\sharp n})$-representation $\delta_1 \mathcal{L}_{((1),\ell)}^*(\mathfrak{n})$ is a free $\mathbb{Z}[Q_{(\lambda,\ell)}^*(S)]$-module of rank 1 for each $\mathfrak{n} \in \mathrm{Obj}(\mathcal{M})$, with generating set given by the tuples $([w_0, w'_0], [w_1, w_2], \dots, [w_{2n-1}, w_{2n}])$ such that $|w_0| + |w'_0| = 1$ if $S = \mathbb{T}$, and the tuples $([w_0], [w_1], \dots, [w_n])$ such that $|w_0| = 1$ if $S = \mathbb{M}$. By expressing $\delta_1 \mathcal{L}_{((1),\ell)}^*([1, \mathrm{id}_{\mathrm{1}\sharp\mathfrak{n}}])$ as a quotient map of $\mathcal{L}_{((1),\ell)}^*(\sigma_1^{-1}) \circ \mathcal{L}_{((1),\ell)}^*([1, \mathrm{id}_{\mathrm{1}\sharp\mathfrak{n}}])$, it follows from equations (3.19) and (3.20)
of Lemma 3.30 and from the description (3.23) of $\mathcal{L}^*_{((1),\ell)}([1, id_{1\otimes n}])$ that $i_1(\delta_1 \mathcal{L}^*_{((1),\ell)})_n$ is an isomorphism for $n \ge 0$. So $\delta_1^2 \mathcal{L}^*_{((1),\ell)} = 0$ and $\kappa_1 \delta_1 \mathcal{L}^*_{((1),\ell)} = 0$, and thus $\mathcal{L}^*_{((1),\ell)}$ is both very strong and weak polynomial, of both strong and weak degrees 1. Furthermore, the part of the proof of Theorem 3.35 showing that the $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module injections $\{(\Delta'_1 \mathcal{L}^*_{(\lambda,\ell)})_n\}_{n \in \text{Obj}(\mathcal{M})}$ (see (3.24)) assemble into a natural transformation in $\text{Fct}(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]\text{-Mod}^\bullet)$ repeats verbatim for $\lambda = (1)$, because Lemmas 3.28 and 3.30 also hold in this case. Then, we deduce from the definition of these injections that $(\Delta_1 \mathcal{L}^*_{((1),\ell)})_n \circ (\Delta'_1 \mathcal{L}^*_{((1),\ell)})_n = \text{id}_{\delta_1 \mathcal{L}^*_{((1),\ell)}(n)}$ for each $n \in \text{Obj}(\mathcal{M})$, and so $\Delta'_1 \mathcal{L}^*_{((1),\ell)}$ is a section of $\Delta_1 \mathcal{L}^*_{((1),\ell)}$ in $\text{Fct}(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{((1),\ell)}(S)]\text{-Mod}^\bullet)$. Since $\kappa_1 \mathcal{L}^*_{((1),\ell)} = 0$ (because $i_1(\mathcal{L}^*_{((1),\ell)})_n$ is clearly injective for each $n \in \mathcal{M}$), the exact sequence (3.1) for $\mathcal{L}^*_{((1),\ell)}$ is a split short exact sequence, and so $\mathcal{L}^*_{((1),\ell)}$ is split polynomial. Furthermore, since $\delta_1 \mathcal{L}^*_{((1),\ell)}(n)$ is a free $\mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]$-module for each $n \in \text{Obj}(\mathcal{M})$ and $\kappa_1 \mathcal{L}^*_{((1),\ell)} = 0$, we deduce that this polynomiality result for $\mathcal{L}^*_{((1),\ell)}$ still holds after any non-zero change of rings operation by using Lemma 3.9 and then repeating verbatim the above argument for the splitting.

We do the inductive step on any fixed $\lambda \vdash k \ge 2$. Namely, we assume that for each $k' \ge 1$ and each $\lambda' \in \{\lambda - k'\}$ (see Notation 3.2), the functor $\mathcal{L}^*_{(\lambda',\ell)}$ is both split and weak polynomial, of both strong and weak degrees $k-k'$, and that this property still holds after any non-zero change of rings operation. Since $\mathcal{SPol}_d(\langle \mathcal{M}, \mathcal{M} \rangle, \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]\text{-Mod}^\bullet)$ is closed under $\tau_1$ and under normal subobjects by Proposition 4.7, this inductive assumption implies that each functor $\tau_1 \mathcal{L}^*_{(\lambda',\ell)}$, and also $(\tau_1 \mathcal{L}^*_{(\lambda',\ell)}(\Gamma))_{[\sigma]}$ in the orientable setting, are split and weak polynomial of both strong and weak degrees $k-k'$, as are the versions of these functors after any non-zero change of rings operation. Now, we deduce from Theorem 3.35 that the translation functor $\tau_1 \mathcal{L}^*_{(\lambda,\ell)}$ is isomorphic to $\mathcal{L}^*_{(\lambda,\ell)} \oplus \delta_1 \mathcal{L}^*_{(\lambda,\ell)}$, where $\delta_1 \mathcal{L}^*_{(\lambda,\ell)}$ is determined by a direct sum of functors (potentially up to a non-zero change of rings, see Convention 3.8) of the form $\tau_1 \mathcal{L}^*_{(\lambda',\ell)}$ with $\lambda' \in \{\lambda-1, \lambda-2\}$, and also $(\tau_1 \mathcal{L}^*_{(\lambda'',\ell)}(\Gamma))_{[\sigma]}$ with $\lambda'' \in \{\lambda-1, \lambda-2\}$ in the orientable setting. Therefore, the functor $\mathcal{L}^*_{(\lambda',\ell)}$ is both very strong and weak polynomial, of both strong and weak degrees $k$. This polynomiality result also holds after applying any non-zero change of rings operation, by using Corollary 3.10 and repeating verbatim the above reasoning, which ends the induction.

Fixing $\ell \in \{1, 2\}$, the same polynomiality results follow for $\mathcal{L}^{*,v}_{(\lambda,\ell)}$ by repeating mutatis mutandis the same arguments, using Theorem 3.36 instead of Theorem 3.35. $\square$

Furthermore, we briefly deal here with the duals of the homological representations of Theorem 3.35. By Corollary 2.14, the MCG($S^{2n}$)-representation $H_k^\partial(C_\lambda(S^{2n} \rtimes I); \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)] \otimes \mathcal{O})$ of §2.3, for $S \in \{\mathbb{T}, \mathbb{M}\}$, is the dual of the MCG($S^{2n}$)-representation $H_k^{\text{BM}}(C_\lambda(S^{2n} \rtimes I); \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)])$. Assigning for each morphism $[m, \text{id}_{m\otimes n}]$ the obvious analogue of the map $\iota_{m,n}$ of §1.2.4 for homology relative to the boundary, these collections of representations extend to functors $\mathcal{L}^{*,\vee}_{(\lambda,\ell)}(\Gamma)$ and $\mathcal{L}^{*,\vee}_{(\lambda,\ell)}(\mathcal{N})$ of the form $\langle \mathcal{M}^\dagger, \mathcal{M}^\dagger \rangle \to \mathbb{Z}[Q^*_{(\lambda,\ell)}(S)]\text{-Mod}$. We may then deduce analogous short exact sequences to those of Theorem 3.36, and Theorem C repeats verbatim for these functors:

**Theorem 4.15** For $\ell \in \{1, 2\}$, the functors $\mathcal{L}^{*,\vee}_{(\lambda,\ell)}(\Gamma)$ and $\mathcal{L}^{*,\vee}_{(\lambda,\ell)}(\mathcal{N})$ are split polynomial and weak polynomial, of both strong and weak degrees $k$.

Finally, as a direct consequence Theorem 4.2, we deduce the following result.

**Corollary 4.16** *Twisted homological stability holds for the mapping class groups of surfaces with coefficients in the homological representation functors of Theorems 3.35 and 3.36.*

## 4.4. Analyticity of a quantum representation

Jackson and Kerler [JK11] introduce a representation $\mathbb{V}$ over the group ring $\mathbb{L} := \mathbb{Z}[\mathfrak{s}^{\pm 1}, \mathfrak{q}^{\pm 1}]$, called the *generic Verma module*, of $U_q(\mathfrak{s}\mathfrak{l}_2)$, the quantum enveloping algebra of the Lie algebra $\mathfrak{s}\mathfrak{l}_2$. Since $U_q(\mathfrak{s}\mathfrak{l}_2)$ is a quasitriangular Hopf algebra, the representation $\mathbb{V}$ comes equipped with an automorphism $S \in \text{Aut}_{\mathbb{L}}(\mathbb{V} \otimes \mathbb{V})$. This induces a $\mathbf{B}_n$-representation on $\mathbb{V}^{\otimes n}$ given by sending $\sigma_i \in \mathbf{B}_n$ to $\text{id}_{i-1} \otimes S \otimes \text{id}_{i-n-i-1}$, which we call the *Verma module representation*; see [JK11, §1]. For $k \ge 0$, the *weight space* $V_{n,k} \subseteq \mathbb{V}^{\otimes n}$ is the eigenspace of the action of a certain generator $K \in U_q(\mathfrak{s}\mathfrak{l}_2)$ corresponding to the eigenvalue $\mathfrak{s}^n \mathfrak{q}^{-2k}$. The $\mathbf{B}_n$-action on $\mathbb{V}^{\otimes n}$ restricts to a sub-$\mathbf{B}_n$-representation on $V_{n,k}$ for each $k \ge 0$, called the *quantum representation* of $\mathbf{B}_n$ of *weight* $k$. This
provides a decomposition of the Verma module representation via the $\mathbf{B}_n$-equivariant isomorphism:

$$ V^{\otimes n} \cong \bigoplus_{k \ge 0} V_{n,k}. \qquad (4.3) $$

The relation between the variables $\mathfrak{s}$ and $\mathfrak{q}$ and the generators $\mathfrak{q}$ and $\mathfrak{t}$ of $Q_{((2),2)}(\mathbb{D}) = \mathbb{Z}^2 = \mathbb{Z}[\mathfrak{q}, \mathfrak{t}]$ (defining the representation $\mathfrak{L}\mathfrak{B}_{((2),2)}(\mathfrak{n})$) is given by the ring homomorphism $\Theta: \mathbb{K} := \mathbb{Z}[\mathfrak{q}^{\pm 1}, \mathfrak{t}^{\pm 1}] \to \mathbb{L}$ defined by $(\mathfrak{q}, \mathfrak{t}) \mapsto (\mathfrak{s}^2, -\mathfrak{q}^{-2})$. (We note as a warning to the reader that the notation in the literature is not consistent; in particular [JK11] and [Mar22] use different notation from each other and from the notation used in this section, which is instead consistent with the notation of [Big01].) In particular, $\mathbb{L}$ is a left $\mathbb{K}$-module via $\Theta$; the change of rings operation $- \otimes_{\mathbb{K}} \mathbb{L}$ corresponds to adjoining square roots of $\mathfrak{q}$ and $\mathfrak{t}$.

A key relationship between these quantum $\mathbf{B}_n$-representations and the homological representation functors studied in this paper is the following lemma.

**Lemma 4.17** For $n, k \ge 1$ there is an isomorphism of $\mathbf{B}_n$-representations $V_{n,k} \cong \tau_1 \mathfrak{L}\mathfrak{B}_k(\mathfrak{n}) \otimes_{\mathbb{K}} \mathbb{L}$.

*Proof.* Let $D'_n$ denote the closed disc minus $n$ interior points and minus one point on its boundary. An alternative description of $\tau_1 \mathfrak{L}\mathfrak{B}_k(\mathfrak{n})$ is given by the twisted Borel-Moore homology of the space of configurations of $k$ unordered points in $D'_n$, namely the $\mathbf{B}_n$-representation $H_k^{\mathrm{BM}}(C_k(D'_n); \mathbb{Z}[\mathbb{Z}^2])$ over $\mathbb{Z}[\mathbb{Z}^2]$. Gluing $D'_0$ to $D'_n$ so that the two boundary punctures coincide induces an embedding $D'_n \hookrightarrow \mathbb{D}_{1+n}$, which in turn induces an embedding $C_k(D'_n) \hookrightarrow C_k(\mathbb{D}_{1+n})$. This latter embedding defines a (covariant) map on Borel-Moore homology since its image is closed, thus it is a proper map, and Borel-Moore homology is covariantly functorial with respect to proper maps; see [Bre97, Proposition V.4.5]. We also note that the local coefficient system that we consider on $C_k(D'_n)$ is the restriction of the one that we consider on $C_k(\mathbb{D}_{1+n})$. There is therefore a well-defined map

$$ H_k^{\mathrm{BM}}(C_k(D'_n); \mathbb{Z}[\mathbb{Z}^2]) \to \tau_1 \mathfrak{L}\mathfrak{B}_k(\mathfrak{n}) \qquad (4.4) $$

of $\mathbf{B}_n$-representations over $\mathbb{Z}[\mathbb{Z}^2] = \mathbb{K}$. The fact that this map is an isomorphism follows from the evident bijection that it induces on the free bases as $\mathbb{K}$-modules obtained from Theorem 2.1.

We now consider the subspace $C_k^-(\mathbb{D}_n) \subset C_k(\mathbb{D}_n)$ of all configurations that intersect a particular fixed point on the boundary. In particular, we consider the $\mathbf{B}_n$-representation given by the $\mathbb{K}$-module $H_k^{\mathrm{BM}}(C_k(\mathbb{D}_n), C_k^-(\mathbb{D}_n); \mathbb{K})$, which is introduced in [Mar22, §2]. Since $C_k(\mathbb{D}'_n)$ is an open subspace of $C_k(\mathbb{D}_n)$ with closed complement $C_k^-(\mathbb{D}_n)$, the inclusion $(C_k(\mathbb{D}'_n), \emptyset) \hookrightarrow (C_k(\mathbb{D}_n), C_k^-(\mathbb{D}_n))$ is an open embedding. Relative Borel-Moore homology is contravariantly functorial with respect to open embeddings (since it is the composition of reduced homology with the contravariant functor from locally-compact, Hausdorff spaces and open embeddings to based spaces given by one-point compactification), so we have a map

$$ H_k^{\mathrm{BM}}(C_k(\mathbb{D}_n), C_k^-(\mathbb{D}_n); \mathbb{K}) \to H_k^{\mathrm{BM}}(C_k(\mathbb{D}'_n); \mathbb{K}). \qquad (4.5) $$

This is a map of $\mathbf{B}_n$-representations over $\mathbb{K}$ since the $\mathbf{B}_n$-action (up to homotopy) on $C_k(\mathbb{D}_n)$ preserves its partition into $C_k^-(\mathbb{D}_n)$ and $C_k(\mathbb{D}'_n)$. The fact that this map is an isomorphism follows from the evident bijection that it induces on the free bases as $\mathbb{K}$-modules obtained from Theorem 2.1 for the right-hand side and [Mar22, Prop. 3.6] for the left-hand side (see also [Mar22, Cor. 3.9]).

Now, [Mar22, Th. 1.5] provides an isomorphism

$$ V_{n,k} \cong H_k^{\mathrm{BM}}(C_k(\mathbb{D}_n), C_k^-(\mathbb{D}_n); \mathbb{K}) \otimes_{\mathbb{K}} \mathbb{L} \qquad (4.6) $$

of $\mathbf{B}_n$-representations over $\mathbb{L}$. The claimed isomorphism of the lemma is then the composition of (4.4), (4.5) and (4.6) ( tensoring the first two isomorphisms over $\mathbb{K}$ with $\mathbb{L}$). $\square$

**Corollary 4.18** For each $n \ge 2$, there is an isomorphism of $\mathbf{B}_n$-representations over $\mathbb{L}$

$$ V^{\otimes n} \cong \bigoplus_{k \ge 0} \tau_1 \mathfrak{L}\mathfrak{B}_k(\mathfrak{n}) \otimes_{\mathbb{K}} \mathbb{L}. \qquad (4.7) $$

We may therefore define the **Verma module representation functor $\mathfrak{Ver}$**: $\langle \beta, \beta \rangle \to \mathbb{L}$-Mod to be the colimit $\bigoplus_{k \ge 0} \tau_1 \mathfrak{L}\mathfrak{B}_k \otimes_{\mathbb{K}} \mathbb{L}$. This functor $\mathfrak{Ver}$ is **analytic**, i.e. it is a colimit of polynomial functors, and **exponential**, i.e. it is a strong monoidal functor $(\langle \beta, \beta \rangle, \mathfrak{s}, \mathfrak{t}, 0) \to (\mathbb{L}$-Mod, $\otimes, \mathbb{L}$). However, the functor $\mathfrak{Ver}$ is **not polynomial**.
*Proof.* The isomorphisms (4.7) follow directly from Lemma 4.17 and the decomposition of the Verma module representation (4.3). The analyticity of the functor $\mathcal{V}\mathcal{er}$ follows from its definition and Corollary B. We deduce from Theorem 3.17 (using Corollary 3.10) that $\delta_1^m \mathcal{V}\mathcal{er} \cong \bigoplus_{k \ge 0} \tau_1^{m+1} \mathfrak{L}\mathfrak{B}_k \otimes_K \mathbb{L}$ for all $m \ge 1$. Hence there is a natural embedding $\mathcal{V}\mathcal{er} \hookrightarrow \delta_1^m \mathcal{V}\mathcal{er}$ for all $m \ge 1$, which proves that the functor $\mathcal{V}\mathcal{er}$ is not polynomial. That it is an exponential functor straightforwardly follows from the isomorphism $\mathcal{V}\mathcal{er}(n) \cong \mathbb{V}^{\otimes n}$. $\square$

**Remark 4.19** Analogous arguments to those of Corollary 4.18 may be repeated verbatim for functors for the mapping class groups of surfaces extending the Magnus representations (see for instance [Sak12, §4] or [Suz05] for the definition of these representations) or the representations constructed from actions on discrete Heisenberg groups introduced by [BPS21].

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Martin Palmer

School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK,

m.d.palmer-anghel@leeds.ac.uk

Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Grivitei, 010702 Bucureşti,
Romania,

mpanghel@imar.ro

Arthur Soulié

Normandie Univ., UNICAEN, CNRS, LMNO, 14000 Caen, France,

artsou@hotmail.fr, arthur.soulie@unicaen.fr