ScalingStacks

Fix $r,n\geq 0$. We define some bimodules $L^{\pm}(r,n)$ and $R^{\pm}(r,n)$ with underlying differential graded module $H_{r+n}$, following §3.1.3 and Proposition 3.1.6.

We endow $L^{+}(r,n)$ (resp. $L^{-}(r,n)$) with a structure of differential graded $(H_{r}\otimes H_{n},H_{r+n})$-bimodule where

• •

  $H_{r+n}$ acts by right multiplication

• •

  $h\in H_{r}$ acts by left multiplication by $h$ (resp. by $f_{n}(h)$)

• •

  $h\in H_{n}$ acts by left multiplication by $f_{r}\circ\iota_{n}(h)$ (resp. by $h$).

We endow $R^{+}(r,n)$ (resp. $R^{-}(r,n)$) with a structure of differential graded $(H_{r+n},H_{r}\otimes H_{n})$-bimodule where

• •

  $H_{r+n}$ acts by left multiplication

• •

  $h\in H_{r}$ acts by right multiplication by $h$ (resp. by $f_{n}(h)$)

• •

  $h\in H_{n}$ acts by right multiplication by $f_{r}\circ\iota_{n}(h)$ (resp. by $h$).

These bimodules coincide with (the nil version of) the bimodules introduced in §3.1.3, after restricting the action of $H_{r}\otimes H_{n}$ to $H_{r}$:

$$L^{\pm}(r,n)=L^{\pm}(I,S)\text{ and }R^{\pm}(r,n)=L^{\pm}(S,I)\text{ where }S=\{s_{1},\ldots,s_{r+n-1}\}\text{ and }I=\{s_{1},\ldots,s_{r-1}\}.$$

Given $m\geq 0$, we denote by $w_{m}\in{\mathfrak{S}}_{m}$ the longest element, i.e., $w_{m}(i)=m-i+1$. We have two morphisms of differential graded ${\mathbf{F}}_{2}$-modules (cf Proposition 3.1.6)

$$t_{r+n,r}^{\pm}=t_{S,I}^{\pm}:H_{r+n}\to H_{r}\langle\frac{1}{2}n(2r+n-1)\rangle$$

given by

$$t_{r+n,r}^{+}(T_{w})=\begin{cases}T_{w_{r}w_{r+n}w}&\text{ if }w\in w_{r+n}{\mathfrak{S}}_{r}\\ 0&\text{ otherwise}\end{cases}\text{ and }t_{r+n,r}^{-}(T_{w})=\begin{cases}T_{ww_{r+n}w_{r}}&\text{ if }w\in{\mathfrak{S}}_{r}w_{r+n}\\ 0&\text{ otherwise}\end{cases}$$

It is immediate that there is an isomorphism of differential graded $(H_{r+n},H_{r}\otimes H_{n})$-modules

$$\operatorname{Hom}\nolimits_{H_{r+n}^{\operatorname{opp}\nolimits}}(L^{\pm}(r,n),H_{r+n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R^{\pm}(r,n),f\mapsto f(1)$$

and it follows from Proposition 3.1.6 that there is an isomorphism of differential graded $(H_{r}\otimes H_{n},H_{r+n})$-modules

$$L^{\mp}(r,n)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{H_{r}^{\operatorname{opp}\nolimits}}(R^{\pm}(r,n),H_{r})\langle\frac{1}{2}n(2r+n-1)\rangle,\ h\mapsto(h^{\prime}\mapsto t_{n+r,r}^{\pm}(hh^{\prime})).$$

We describe now $L^{+}(r,n)$ as a twisted free $(H_{r}\otimes H_{n})$-module.

Consider $E\subset\{1,\ldots,r+n\}$ with $|E|=r$. Let $w_{E}\in{\mathfrak{S}}_{r+n}$ be the permutation such that $w_{E}(E)=\{1,\ldots,r\}$ and the restrictions of $w_{E}$ to $E$ and to $\{1,\ldots,r+n\}\setminus E$ are increasing. If $E=\{i_{1}<\cdots<i_{r}\}$, then we have a reduced decomposition

$$w_{E}=(s_{r}\cdots s_{i_{r}-1})(s_{r-1}\cdots s_{i_{r-1}-1})\cdots(s_{2}\cdots s_{i_{2}-1})(s_{1}\cdots s_{i_{1}-1})$$

and

$$\tilde{L}(w_{E})=\coprod_{b=1}^{r}\bigl((\{1,\ldots,i_{b}-1\}\setminus\{i_{1},\ldots,i_{b-1}\})\times\{i_{b}\}\bigr).$$

There is a bijection

$$\beta:{\mathfrak{S}}_{r}\times{\mathfrak{S}}_{n}\times\{E\subset\{1,\ldots,r+n\}\ |\ |E|=r\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathfrak{S}}_{r+n},(v,v^{\prime},E)\mapsto vf_{r}(v^{\prime})w_{E}$$

where $f_{r}(v^{\prime})\in{\mathfrak{S}}_{n+r}$ is given by $f_{r}(v^{\prime})(i)=i$ for $i\leq r$ and $f_{r}(v^{\prime})(r+i)=r+v^{\prime}(i)$ for $1\leq i\leq n$. We have $\ell(\beta(v,v^{\prime},E))=\ell(v)+\ell(v^{\prime})+\ell(w_{E})$.

Given $(a,i_{b})\in\tilde{L}(w_{E})$, we define $v(E,a,b)\in{\mathfrak{S}}_{r}$ and $v^{\prime}(E,a,b)\in{\mathfrak{S}}_{n}$ as follows. Let $b^{\prime}\in\{1,\ldots,r\}$ be minimal such that $a<i_{b^{\prime}}$. We define $v(E,a,b)$ to be the cycle $(b,b-1,\ldots,b^{\prime})$ and $v^{\prime}(E,a,b)$ to be the cycle $(a-b^{\prime}+1,a-b^{\prime}+2,\ldots,i_{b}-b)$. We have

$$w_{E}s_{a,i_{b}}=v(E,a,b)f_{r}(v^{\prime}(E,a,b))w_{(E\cup\{a\})\setminus\{i_{b}\}}$$

and $\ell(w_{E})-\ell(w_{(E\cup\{a\})\setminus\{i_{b}\}})=i_{b}-a$.

Given $m\geq 1$, we define a free differential $(H_{r}\otimes H_{n})$-module

$$V_{m}=\bigoplus_{E\subset\{1,\ldots,r+n\},\ |E|=r,\ \ell(w_{E})=m-1}(H_{r}\otimes H_{n})b_{E}.$$

Given $m^{\prime}<m$, we define $f_{m^{\prime},m}:V_{m}\to V_{m^{\prime}}$ as the morphism of $(H_{r}\otimes H_{n})$-modules given by

$$b_{E}\mapsto\sum_{\begin{subarray}{c}i\in E,\ j\in\{1,\ldots,r+n\}\setminus E\\ i-j=m-m^{\prime}\end{subarray}}(T_{v(E,j,i)}\otimes T_{v^{\prime}(E,j,i)})b_{(E\cup\{j\})\setminus\{i\}}.$$

We will show below (Lemma 6.1.3) that $d(f_{m^{\prime},m})=\sum_{m>m^{\prime\prime}>m^{\prime}}f_{m^{\prime}m^{\prime\prime}}\circ f_{m^{\prime\prime}m}$. We denote by $V$ the differential $(H_{r}\otimes H_{n})$-module obtained as the corresponding twisted object $[\bigoplus V_{m},(f_{m^{\prime}m})]$ (cf §2.1.3). We have $V=\bigoplus_{m}V_{m}$ as a $(H_{r}\otimes H_{n})$-module and $d_{V}=\sum_{m}d_{V_{m}}+\sum_{m,m^{\prime}}f_{m^{\prime},m}$.

There is a dual version of Lemma 6.1.3. In particular, there is a decomposition of right $(H_{r}\otimes H_{n})$-modules

$$R^{+}(r,n)=\bigoplus_{E\subset\{1,\ldots,r+n\},\ |E|=r}T_{w_{E}^{-1}}(H_{r}\otimes f_{r}(H_{n}))$$

There is a “left” $2$-representation on ${\mathcal{U}}$

$$\Upsilon^{-}:{\mathcal{U}}\to\operatorname{End}\nolimits({\mathcal{U}}),\ e^{n}\mapsto e^{n}\otimes-$$

and a “right” $2$-representation on ${\mathcal{U}}$

$$\Upsilon^{+}:{\mathcal{U}}\xrightarrow[\sim]{\mathrm{rev}}{\mathcal{U}}^{\mathrm{rev}}\xrightarrow{e^{n}\mapsto-\otimes e^{n}}\operatorname{End}\nolimits({\mathcal{U}}).$$

The bimodule $2$-representation $L^{\pm}$ associated to $\Upsilon^{\pm}$ is given by

$$L^{\pm}(e^{r},e^{s},e^{n})=\delta_{s,r+n}L^{\pm}(r,n)$$

and it is left and right finite. Its left dual is isomorphic to the bimodule $2$-representation $R^{\pm}$ given by

$$R^{\pm}(e^{s},e^{r},e^{n})=\delta_{s,r+n}R^{\pm}(r,n)$$

while its right dual is isomorphic to $R^{\mp}\langle-\frac{1}{2}n(2r+n-1)\rangle$ (note that the action of ${\mathcal{U}}$ on the duals is obtained from the natural action of ${\mathcal{U}}^{\mathrm{rev}{\operatorname{opp}\nolimits}}$ by applying the isomorphism $\mathrm{rev}\circ{\operatorname{opp}\nolimits}$).

The $2$-representations $\Upsilon^{+}$ and $\Upsilon^{-}$ commute strictly. Let us describe this in terms of bimodules.

We consider the bimodule $2$-representations $E_{1}=\bigoplus_{s\geq 0}L^{-}(s,1)$ and $E_{2}=\bigoplus_{s\geq 0}L^{+}(s,1)$ as above. There is a canonical isomorphism $E_{1}^{\vee}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\bigoplus_{s\geq 0}R^{-}(s,1)$ and we identify those bimodules.

Define $\sigma:E_{2}E_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{1}E_{2}$ as the isomorphism such that for $s\geq 1$, the following diagram of morphism of $(H_{s-1},H_{s+1})$-bimodules is commutative:

Note that the left action of $a\in H_{s-1}$ on $H_{s+1}$ is given by left multiplication by $f_{1}(a)$. It is immediate to check that the diagrams (4.3.1) commute.

As in (5.3.1), the morphism $\sigma$ gives a morphism of functors

$$\lambda:R^{-}(-_{1},-,e)\otimes L^{+}(-,-_{2},e)\to L^{+}(-_{1},-,e)\otimes R^{-}(-,-_{2},e)$$

where $\lambda(e^{s},e^{s})$ is given by the following morphism of differential graded $(H_{s},H_{s})$-bimodules

The morphism

$$(R^{-}(-,-,e)\lambda L^{+}(-,-,e))\circ(R^{-}(-,-,e)^{2}\tau-\tau L^{+}(-,-,e)^{2}):\\ R^{-}(-,-,e)^{2}L^{+}(-,-,e)^{2}\to(R^{-}(-,-,e)L^{+}(-,-,e))^{2}\langle-1\rangle$$

is on $(e^{s},e^{s})$ the morphism of differential graded $(H_{s},H_{s})$-bimodules

$$R^{-}(s-1,1)\otimes_{H_{s-1}}R^{-}(s-2,1)\otimes_{H_{s-2}}L^{+}(s-2,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\to\\ R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\otimes_{H_{s}}R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)\langle-1\rangle$$

given by

$$1\otimes 1\otimes 1\otimes 1\mapsto T_{1}\otimes 1\otimes 1\otimes 1-1\otimes 1\otimes 1\otimes T_{s-1}.$$

Given $s\geq 1$, let $M_{s}=R^{-}(s-1,1)\otimes_{H_{s-1}}L^{+}(s-1,1)$, a differential graded $(H_{s},H_{s})$-bimodule. When $s\geq 2$, we define $\kappa=1\otimes 1\otimes 1\otimes T_{s-1}-T_{1}\otimes 1\otimes 1\otimes 1\in M_{s}\otimes_{H_{s}}M_{s}$. We put $\kappa=0$ when $s=1$. We put $M_{0}=0$.

Let ${\mathcal{H}}^{+}$ be the differential graded pointed category with set of objects ${\mathbf{Z}}_{\geq 0}$ and $\operatorname{Hom}\nolimits_{{\mathcal{H}}^{+}}(m,n)=\delta_{mn}\hat{{\mathfrak{S}}}_{n}^{+,\mathrm{nil}}$. Lemma 6.1.5 has the following consequence.

Since $\sigma$ is an isomorphism, we have a diagonal bimodule $2$-representation on $\Delta^{\prime}_{\lambda}({\mathcal{U}}^{\bullet})$ (cf §5.3.3). Via the isomorphism of Theorem 6.1.6, this corresponds to the bimodule $2$-representation on ${\mathcal{H}}^{+}$ defined as follows. Define a differential graded right $\hat{H}_{n}^{+}$-module

$$E_{n}={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 22.47226pt\hbox{{\hbox{\kern-22.47226pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.73114pt\hbox{$\textstyle{\hat{H}_{n}^{+}[1]\oplus\hat{H}_{n}^{+}}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-9.46863pt\raise 22.8116pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h\mapsto ch}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 14.2263pt\raise 11.38104pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}}$$

We define a left action of $\hat{H}_{n-1}^{+}$ on $E_{n}$ as follows:

$$T_{i}\text{ acts by }\left(\begin{matrix}T_{i}&0\\ 0&T_{i+1}\end{matrix}\right)\text{ for }1\leq i\leq n-2\text{ and }c\text{ acts by }\left(\begin{matrix}cT_{n-1}&1\\ 0&T_{1}c\end{matrix}\right).$$

This defines a structure of differential graded $(\hat{H}_{n-1}^{+},\hat{H}_{n}^{+})$-bimodule on $E_{n}$.

Note that setting $E=0$ corresponds to inverting $c$: this turns ${\mathcal{H}}^{+}$ into the differential graded pointed category with same objects and with $\operatorname{Hom}\nolimits_{{\mathcal{H}}}(m,n)=\delta_{mn}\hat{{\mathfrak{S}}}_{n}^{\mathrm{nil}}$.