ScalingStacks

Consider a differential category 𝒲{\mathcal{W}} endowed with two actions of 𝒰{\mathcal{U}} given by (E1,Ο„1)(E_{1},\tau_{1}) and (E2,Ο„2)(E_{2},\tau_{2}) and a closed morphism of functors Οƒ:E2​E1β†’E1​E2\sigma:E_{2}E_{1}\to E_{1}E_{2} such that the following diagrams commute:

(4.3.1) E22​E1\textstyle{E_{2}^{2}E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​σ\scriptstyle{E_{2}\sigma}Ο„2​E1\scriptstyle{\tau_{2}E_{1}}E2​E1​E2\textstyle{E_{2}E_{1}E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ​E2\scriptstyle{\sigma E_{2}}E1​E22\textstyle{E_{1}E_{2}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​τ2\scriptstyle{E_{1}\tau_{2}}E22​E1\textstyle{E_{2}^{2}E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E2​σ\scriptstyle{E_{2}\sigma}E2​E1​E2\textstyle{E_{2}E_{1}E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ​E2\scriptstyle{\sigma E_{2}}E1​E22\textstyle{E_{1}E_{2}^{2}}     E2​E12\textstyle{E_{2}E_{1}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ​E1\scriptstyle{\sigma E_{1}}E2​τ1\scriptstyle{E_{2}\tau_{1}}E1​E2​E1\textstyle{E_{1}E_{2}E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​σ\scriptstyle{E_{1}\sigma}E12​E2\textstyle{E_{1}^{2}E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ο„1​E2\scriptstyle{\tau_{1}E_{2}}E2​E12\textstyle{E_{2}E_{1}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ​E1\scriptstyle{\sigma E_{1}}E1​E2​E1\textstyle{E_{1}E_{2}E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1​σ\scriptstyle{E_{1}\sigma}E12​E2\textstyle{E_{1}^{2}E_{2}}

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2