ScalingStacks

4.4.2. Adjoint

We assume F1F_{1} has a right adjoint E1E_{1} and denote by ε1\varepsilon_{1} and η1\eta_{1} the counit and unit of the adjunction. We denote by τ1\tau_{1} the endomorphism of E12E_{1}^{2} corresponding by adjunction to the endomorphism τ1\tau_{1} of F12F_{1}^{2}. The pair (E1,τ1)(E_{1},\tau_{1}) provides an action of 𝒰{\mathcal{U}} on 𝒲{\mathcal{W}}.

0P68

Remark 4.4.4. The maps η1\eta_{1}, ε1\varepsilon_{1}, the relations they satisfy, and λ\lambda, σ\sigma and ρ\rho are described graphically as:

[Uncaptioned image]

We denote by σ\sigma the composition

(4.4.1) σ:E2​E1→η1​E2​E1E1​F1​E2​E1→E1​λ​E2E1​E2​F1​E1→E1​E2​ε1E1​E2\sigma:E_{2}E_{1}\xrightarrow{\eta_{1}E_{2}E_{1}}E_{1}F_{1}E_{2}E_{1}\xrightarrow{E_{1}\lambda E_{2}}E_{1}E_{2}F_{1}E_{1}\xrightarrow{E_{1}E_{2}\varepsilon_{1}}E_{1}E_{2}

and by ρ\rho the composition

(4.4.2) ρ:F1​E1→F1​E1​η1F1​E12​F1→F1​τ1​F1F1​E12​F1→ε1​E1​F1E1​F1.\rho:F_{1}E_{1}\xrightarrow{F_{1}E_{1}\eta_{1}}F_{1}E_{1}^{2}F_{1}\xrightarrow{F_{1}\tau_{1}F_{1}}F_{1}E_{1}^{2}F_{1}\xrightarrow{\varepsilon_{1}E_{1}F_{1}}E_{1}F_{1}.

The diagram (4.3.1) is commutative.

0P69

Lemma 4.4.5. We have

E1​λ∘ρ​E2∘F1​σ=σ​F1∘E2​ρ∘λ​E1​ and ​ρ​F1∘F1​ρ∘τ1​E1=E1​τ1∘ρ​F1∘F1​ρ.E_{1}\lambda\circ\rho E_{2}\circ F_{1}\sigma=\sigma F_{1}\circ E_{2}\rho\circ\lambda E_{1}\text{ and }\rho F_{1}\circ F_{1}\rho\circ\tau_{1}E_{1}=E_{1}\tau_{1}\circ\rho F_{1}\circ F_{1}\rho.
0P6A

Proof. We have

E1​λ∘ρ​E2∘F1​σ=E_{1}\lambda\circ\rho E_{2}\circ F_{1}\sigma=
=E1​E2​F1​ε1∘E1​λ​F1​E1∘F1​E12​F1​λ​E1∘F1​τ1​F12​E2​E1∘F1​E1​η1​F1​E2​E1∘F1​η1​E2​E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ F_{1}E_{1}^{2}F_{1}\lambda E_{1}\circ F_{1}\tau_{1}F_{1}^{2}E_{2}E_{1}\circ F_{1}E_{1}\eta_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2}
=E1​E2​F1​ε1∘E1​λ​F1​E1∘F1​E12​F1​λ​E1∘F1​E12​τ1​E2​E1∘F1​E1​η1​F1​E2​E1∘F1​η1​E2​E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ F_{1}E_{1}^{2}F_{1}\lambda E_{1}\circ F_{1}E_{1}^{2}\tau_{1}E_{2}E_{1}\circ F_{1}E_{1}\eta_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2}
=E1​E2​F1​ε1∘E1​λ​F1​E1∘E1​F1​λ​E1∘E1​τ1​E2​E1∘η1​F1​E2​E1∘ε1​F1​E2​E1∘F1​η1​E2​E2\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ E_{1}\tau_{1}E_{2}E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}\circ\varepsilon_{1}F_{1}E_{2}E_{1}\circ F_{1}\eta_{1}E_{2}E_{2}
=E1​E2​F1​ε1∘E1​λ​F1​E1∘E1​F1​λ​E1∘E1​τ1​E2​E1∘η1​F1​E2​E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ E_{1}\tau_{1}E_{2}E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}
=E1​E2​F1​ε1∘E1​E2​τ1​E1∘E1​λ​F1​E1∘E1​F1​λ​E1∘η1​F1​E2​E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}\tau_{1}E_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}
=E1​E2​F1​ε1∘E1​E2​F1​ε1​E1​F1∘E1​E2​F12​E1​η1∘E1​E2​τ1​E1∘E1​λ​F1​E1∘E1​F1​λ​E1∘η1​F1​E2​E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}E_{2}\tau_{1}E_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}
=E1​E2​F1​ε1∘E1​E2​F1​ε1​E1​F1∘E1​E2​τ1​E12​F1∘E1​E2​F12​E1​η1∘E1​λ​F1​E1∘E1​F1​λ​E1∘η1​F1​E2​E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}\tau_{1}E_{1}^{2}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}
=E1​E2​F1​ε1∘E1​E2​F1​ε1​E1​F1∘E1​E2​F12​τ1​F1∘E1​E2​F12​E1​η1∘E1​λ​F1​E1∘E1​F1​λ​E1∘η1​F1​E2​E1\displaystyle=E_{1}E_{2}F_{1}\varepsilon_{1}\circ E_{1}E_{2}F_{1}\varepsilon_{1}E_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}\tau_{1}F_{1}\circ E_{1}E_{2}F_{1}^{2}E_{1}\eta_{1}\circ E_{1}\lambda F_{1}E_{1}\circ E_{1}F_{1}\lambda E_{1}\circ\eta_{1}F_{1}E_{2}E_{1}
=σ​F1∘E2​ρ∘λ​E1.\displaystyle=\sigma F_{1}\circ E_{2}\rho\circ\lambda E_{1}.

We have

ρ​F1∘F1​ρ∘τ1​E1\displaystyle\rho F_{1}\circ F_{1}\rho\circ\tau_{1}E_{1} =ε1​E1​F12∘F1​ε1​E12​F12∘τ1​E13​F12∘F12​E1​τ1​F12∘F12​E12​η1​F1∘F12​τ1​F1∘F12​E1​η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ\tau_{1}E_{1}^{3}F_{1}^{2}\circ F_{1}^{2}E_{1}\tau_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}\tau_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1}
=ε1​E1​F12∘F1​ε1​E12​F12∘F12​τ1​E1​F12∘F12​E1​τ1​F12∘F12​E12​η1​F1∘F12​τ1​F1∘F12​E1​η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}\tau_{1}E_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}\tau_{1}F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}\tau_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1}
=ε1​E1​F12∘F1​ε1​E12​F12∘F12​(τ1​E1∘E1​τ1∘τ1​E1)​F12∘F12​E12​η1​F1∘F12​E1​η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(\tau_{1}E_{1}\circ E_{1}\tau_{1}\circ\tau_{1}E_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1}
=ε1​E1​F12∘F1​ε1​E12​F12∘F12​(E1​τ1∘τ1​E1∘E1​τ1)​F12∘F12​E12​η1​F1∘F12​E1​η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(E_{1}\tau_{1}\circ\tau_{1}E_{1}\circ E_{1}\tau_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1}
=ε1​E1​F12∘F1​ε1​E12​F12∘F12​(E1​τ1∘τ1​E1)​F12∘F12​E13​τ1∘F12​E12​η1​F1∘F12​E1​η1\displaystyle=\varepsilon_{1}E_{1}F_{1}^{2}\circ F_{1}\varepsilon_{1}E_{1}^{2}F_{1}^{2}\circ F_{1}^{2}(E_{1}\tau_{1}\circ\tau_{1}E_{1})F_{1}^{2}\circ F_{1}^{2}E_{1}^{3}\tau_{1}\circ F_{1}^{2}E_{1}^{2}\eta_{1}F_{1}\circ F_{1}^{2}E_{1}\eta_{1}
=E1​τ1∘ρ​F1∘F1​ρ.\displaystyle=E_{1}\tau_{1}\circ\rho F_{1}\circ F_{1}\rho.

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2