5.3.2. Left dual
Let B B 2 2 ( E 1 , τ 1 ) (E_{1},\tau_{1}) ( E 2 , τ 2 ) (E_{2},\tau_{2})
We consider the data
of σ ∈ Z Hom ( E 2 E 1 , E 1 E 2 ) \sigma\in Z\operatorname{Hom}\nolimits(E_{2}E_{1},E_{1}E_{2}) 4.3.1
We define
(5.3.1) λ : E 1 ∨ E 2 → ∙ η 1 E 1 ∨ E 2 E 1 E 1 ∨ → E 1 ∨ σ E 1 ∨ E 1 ∨ E 1 E 2 E 1 ∨ → ε 1 ∙ E 2 E 1 ∨ . \lambda:E_{1}^{\vee}E_{2}\xrightarrow{\bullet\eta_{1}}E_{1}^{\vee}E_{2}E_{1}E_{1}^{\vee}\xrightarrow{E_{1}^{\vee}\sigma E_{1}^{\vee}}E_{1}^{\vee}E_{1}E_{2}E_{1}^{\vee}\xrightarrow{\varepsilon_{1}\bullet}E_{2}E_{1}^{\vee}.
Let A = Δ σ ( B ) = Δ λ ′ ( B ) A=\Delta_{\sigma}(B)=\Delta^{\prime}_{\lambda}(B) T B ( E 1 ∨ E 2 ) T_{B}(E_{1}^{\vee}E_{2})
( E 1 ∨ ) 2 E 2 2 → τ 1 E 2 2 − ( E 1 ∨ ) 2 τ 2 ( E 1 ∨ ) 2 E 2 2 → E 1 ∨ λ E 2 ( E 1 ∨ E 2 ) 2 . (E_{1}^{\vee})^{2}E_{2}^{2}\xrightarrow{\tau_{1}E_{2}^{2}-(E_{1}^{\vee})^{2}\tau_{2}}(E_{1}^{\vee})^{2}E_{2}^{2}\xrightarrow{E_{1}^{\vee}\lambda E_{2}}(E_{1}^{\vee}E_{2})^{2}.
The algebra A A A 0 = B A^{0}=B A 1 = E 1 ∨ E 2 A^{1}=E_{1}^{\vee}E_{2}
Let L L B B A A L L B B B B ς : E 1 ∨ E 2 ⊗ B L → L \varsigma:E_{1}^{\vee}E_{2}\otimes_{B}L\to L d ( ς ) = 0 d(\varsigma)=0
(5.3.2) ( E 1 ∨ ) 2 E 2 2 L \textstyle{(E_{1}^{\vee})^{2}E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 ∨ λ E 2 \scriptstyle{E_{1}^{\vee}\lambda E_{2}} ( E 1 ∨ E 2 ) 2 L \textstyle{(E_{1}^{\vee}E_{2})^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 ∨ E 2 ς \scriptstyle{E_{1}^{\vee}E_{2}\varsigma} E 1 ∨ E 2 L \textstyle{E_{1}^{\vee}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ς \scriptstyle{\varsigma} ( E 1 ∨ ) 2 E 2 2 L \textstyle{(E_{1}^{\vee})^{2}E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} τ 1 E 2 2 \scriptstyle{\tau_{1}E_{2}^{2}} ( E 1 ∨ ) 2 τ 2 \scriptstyle{(E_{1}^{\vee})^{2}\tau_{2}} L \textstyle{L} ( E 1 ∨ ) 2 E 2 2 L \textstyle{(E_{1}^{\vee})^{2}E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 ∨ λ E 2 \scriptstyle{E_{1}^{\vee}\lambda E_{2}} ( E 1 ∨ E 2 ) 2 L \textstyle{(E_{1}^{\vee}E_{2})^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 ∨ E 2 ς \scriptstyle{E_{1}^{\vee}E_{2}\varsigma} E 1 ∨ E 2 L \textstyle{E_{1}^{\vee}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ς \scriptstyle{\varsigma}
This gives us an identification (isomorphism of categories)
between differential A A B B L L ς \varsigma
Consider the adjunction isomorphism
ϕ : Hom B ( E 2 L , E 1 L ) → ∼ Hom B ( E 1 ∨ E 2 L , L ) \phi:\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{B}(E_{1}^{\vee}E_{2}L,L)
Let π ∈ Z Hom B ( E 2 L , E 1 L ) \pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L) ς = ϕ ( π ) ∈ Z Hom B ( E 1 ∨ E 2 L , L ) \varsigma=\phi(\pi)\in Z\operatorname{Hom}\nolimits_{B}(E_{1}^{\vee}E_{2}L,L) 5.3.2
(5.3.3) E 2 2 L \textstyle{E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 2 π \scriptstyle{E_{2}\pi} τ 2 \scriptstyle{\tau_{2}} E 2 E 1 L \textstyle{E_{2}E_{1}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} σ \scriptstyle{\sigma} E 1 E 2 L \textstyle{E_{1}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 π \scriptstyle{E_{1}\pi} E 1 2 L \textstyle{E_{1}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} τ 1 \scriptstyle{\tau_{1}} E 2 2 L \textstyle{E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 2 π \scriptstyle{E_{2}\pi} E 2 E 1 L \textstyle{E_{2}E_{1}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} σ \scriptstyle{\sigma} E 1 E 2 L \textstyle{E_{1}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 π \scriptstyle{E_{1}\pi} E 1 2 L \textstyle{E_{1}^{2}L}
This gives us an identification (isomorphism of categories)
between differential A A [ L , π ] [L,\pi] L L B B π ∈ Z Hom B ( E 2 L , E 1 L ) \pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L) 5.3.3
0P6V
Lemma 5.3.2 . The construction ( m , π ) ↦ [ m , π ] (m,\pi)\mapsto[m,\pi] Φ : Δ σ ( B − diff ) → ( Δ σ B ) − diff \Phi:\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\to(\Delta_{\sigma}B)\operatorname{\!-diff}\nolimits
We will show that the structure of 2 2 Δ σ ( B − diff ) \Delta_{\sigma}(B\operatorname{\!-diff}\nolimits) 2 2 Δ σ B \Delta_{\sigma}B σ \sigma
![[Uncaptioned image]](/assets/author-raster/W000055/914151a932f4ff31e44290b6d13f91216b47000fe9fbc34b583ee1e5462ecb06.png)