ScalingStacks

Let $A$ be a differential algebra.

Consider a $2$-representation on $A$. Note that $E\otimes_{A}-$ is a differential endofunctor of $A\operatorname{\!-diff}\nolimits$, and $\tau$ defines an endomorphism of $(E\otimes_{A}-)^{2}$. This gives a structure of $2$-representation on $A\operatorname{\!-diff}\nolimits$. It restricts to a $2$-representation on $(\bar{A})^{i}$ if $E$ is strictly perfect as a differential $A$-module.

Note that there is a morphism of differential algebras

Let $A^{\prime}$ be another differential algebra with a $2$-representation $(E^{\prime},\tau^{\prime})$. We define a morphism of $2$-representations from $(A,E,\tau)$ to $(A^{\prime},E^{\prime},\tau^{\prime})$ to be an $(A^{\prime},A)$-bimodule $P$ together with a closed isomorphism of $(A^{\prime},A)$-bimodules $\varphi:P\otimes_{A}E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\otimes_{A^{\prime}}P$ such that

Note that such a pair $(P,\varphi)$ gives rise to a morphism of $2$-representations $(P\otimes_{A}-,\varphi):(A\operatorname{\!-diff}\nolimits,E\otimes_{A}-,\tau)\to(A^{\prime}\operatorname{\!-diff}\nolimits,E^{\prime}\otimes_{A^{\prime}}-,\tau^{\prime})$.

We obtain a differential $2$-category of $2$-representations on differential algebras.

The opposite $2$-representation is the data $(A^{\prime},E^{\prime},\tau^{\prime})$ where $A^{\prime}=A^{\operatorname{opp}\nolimits}$, $E^{\prime}=E$ and $\tau^{\prime}=\tau$. Note that $(A,E,\tau)$ coincides with its double dual.

Assume now the $2$-representation is right finite. We have two morphisms of $(A,A)$-bimodules $\eta:A\to E\otimes_{A}E^{\vee}$ and $\varepsilon:E^{\vee}\otimes_{A}E\to A$ (unit and counit of adjunction). We have a morphism of $(A,A)$-bimodules $\rho:E^{\vee}E\to EE^{\vee}$ defined as the composition

There is a canonical isomorphism of differential algebras $\operatorname{End}\nolimits(E^{2})^{\operatorname{opp}\nolimits}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits((E^{\vee})^{2})$ and we still denote by $\tau$ the endomorphism of $(E^{\vee})^{2}$ corresponding to $\tau$.

We define the left dual $2$-representation on $A$ with the bimodule $E^{\vee}$ and the endomorphism $\tau$.

Let $B$ be a differential algebra endowed with two $2$-representations $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ together with a closed morphism $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the diagrams (4.2.1) commute.

We define the algebra $A=\Delta^{\prime}_{\lambda}(B)$ as the quotient of the tensor algebra $T_{B}(F_{1}E_{2})$ by the two-sided ideal generated by the image of the composition

We have $A^{0}=B$ and $A^{1}=F_{1}E_{2}$.

Let $B^{\prime}$ be a differential algebra endowed with two $2$-representations $(F^{\prime}_{1},\tau^{\prime}_{1})$ and $(E^{\prime}_{2},\tau^{\prime}_{2})$ together with a closed morphism $\lambda^{\prime}:F^{\prime}_{1}E^{\prime}_{2}\to E^{\prime}_{2}F^{\prime}_{1}$ such that the analogs of the diagrams (4.2.1) commute. Let $A^{\prime}=\Delta^{\prime}_{\lambda^{\prime}}(B^{\prime})$. Let $P$ be a $(B^{\prime},B)$-bimodule and $\varphi_{1}:PF_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F^{\prime}_{1}P$ and $\varphi_{2}:PE_{2}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}_{2}P$ be two closed isomorphisms of bimodules such that $(P,\varphi_{1})$ and $(P,\varphi_{2})$ are morphisms of $2$-representations and such that

The isomorphism $F^{\prime}_{1}\varphi_{2}\circ\varphi_{1}E_{2}:PF_{1}E_{2}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F^{\prime}_{1}E_{2}^{\prime}P$ induces an isomorphism of $(B^{\prime},B)$-bimodules $f:P\otimes_{B}T_{B}(F_{1}E_{2})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}T_{B^{\prime}}(F^{\prime}_{1}E^{\prime}_{2})\otimes_{B^{\prime}}P$. This isomorphism $f$ endows the right $T_{B}(F_{1}E_{2})$-module $P\otimes_{B}T_{B}(F_{1}E_{2})$ with a commuting left action of $T_{B^{\prime}}(F^{\prime}_{1}E^{\prime}_{2})$. The isomorphism $f$ induces an isomorphism

So, we obtain a structure of $(A^{\prime},A)$-bimodule on $P\otimes_{B}A$.

Let $B$ be a differential algebra endowed with two $2$-representations $(E_{1},\tau_{1})$ and $(E_{2},\tau_{2})$, the first of which is right finite.

We consider the data of $\sigma\in Z\operatorname{Hom}\nolimits(E_{2}E_{1},E_{1}E_{2})$ such that the diagrams (4.3.1) commute.

We define

Let $A=\Delta_{\sigma}(B)=\Delta^{\prime}_{\lambda}(B)$. This is the graded quotient of the tensor algebra $T_{B}(E_{1}^{\vee}E_{2})$ by the ideal generated by the image of the composition

The algebra $A$ is generated by $A^{0}=B$ and $A^{1}=E_{1}^{\vee}E_{2}$.

Let $L$ be a differential $B$-module. The data of a structure of $A$-module on $L$ extending the action of $B$ is the same as the data of a morphism of $B$-modules $\varsigma:E_{1}^{\vee}E_{2}\otimes_{B}L\to L$ such that $d(\varsigma)=0$ and the following diagram commutes

This gives us an identification (isomorphism of categories) between differential $A$-modules and pairs consisting of a differential $B$-module $L$ and a map $\varsigma$ as above.

Consider the adjunction isomorphism

Let $\pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L)$ and let $\varsigma=\phi(\pi)\in Z\operatorname{Hom}\nolimits_{B}(E_{1}^{\vee}E_{2}L,L)$. The commutativity of the diagram (5.3.2) is equivalent to the commutativity of the diagram

This gives us an identification (isomorphism of categories) between differential $A$-modules and pairs $[L,\pi]$ where $L$ is a differential $B$-module, $\pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L)$ and the diagram (5.3.3) commutes. We have obtained the following lemma.

We will show that the structure of $2$-representation on $\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)$ comes from a structure of $2$-representation on $\Delta_{\sigma}B$, when $\sigma$ is invertible.

We define the closed morphism of $(B,A)$-bimodules $u:E_{2}\otimes_{B}A\to E_{1}\otimes_{B}A$ as the adjoint to the multiplication map $E_{1}^{\vee}E_{2}\otimes_{B}A\to A$. We define $E$ as the cone of $u$.

We define a morphism of $(B,A)$-bimodules $v:E_{2}\otimes_{B}E\to E_{1}\otimes_{B}E$ by

Note that the morphism $v$ corresponds, by adjunction, to the morphism $w:E_{1}^{\vee}E_{2}\otimes_{B}E\to E$ defined as follows

We assume now that $\sigma$ is invertible. We define $\tau$ an endomorphism of $(B,A)$-bimodules of $E_{2}^{2}\otimes_{B}A\oplus E_{2}E_{1}\otimes_{B}A\oplus E_{1}E_{2}\otimes_{B}A\oplus E_{1}^{2}\otimes_{B}A$ by

Let $A_{1}$ and $A_{2}$ be two differential algebras equipped with structures of $2$-representations $(E_{i},\tau_{i})$, $i=1,2$.

Let $B=A_{1}\otimes A_{2}$. It is endowed with commuting $2$-representations $(E_{1}\otimes A_{2},\tau_{1}\otimes 1)$ and $(A_{1}\otimes E_{2},1\otimes\tau_{2})$: the isomorphism $\sigma$ is induced by the swap map $E_{2}\otimes E_{1}\xrightarrow{\sim}E_{1}\otimes E_{2},\ a_{2}\otimes a_{1}\mapsto a_{1}\otimes a_{2}$. The tensor product identifies $(A_{1}\operatorname{\!-diff}\nolimits)\otimes(A_{2}\operatorname{\!-diff}\nolimits)$ with a full subcategory of $B\operatorname{\!-diff}\nolimits$.

Assume $E_{1}$ is right finite. The map $\lambda$ is an isomorphism. We put $A_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}A_{2}=\Delta^{\prime}_{\lambda}(B)$. It is the quotient of the tensor algebra $T_{A_{1}\otimes A_{2}}(E_{1}^{\vee}\otimes E_{2})$ by the ideal generated by $p\tau_{2}(q)-\tau_{1}(p)q$ for $p\in(E_{1}^{\vee})^{\otimes 2}$ and $q\in(E_{2})^{\otimes 2}$. The underlying differential module is

The multiplication is defined by

We have

The right action of $A$ on $E$ is given by right multiplication, while the left action of $E_{1}^{\vee}\otimes E_{2}$ on $A_{1}\otimes E_{2}\oplus E_{1}\otimes A_{2}\subset E$ is given by