8.1.3. 2 2
Let f : Z → Z ′ f:Z\to Z^{\prime} ξ \xi ( Z , M ) (Z,M) f ∘ ξ f\circ\xi ( Z ′ , f ( M ) ) (Z^{\prime},f(M))
Assume that | f − 1 ( f ( z ) ) | = 1 |f^{-1}(f(z))|=1 z ∈ M z\in M M f M_{f} ( 𝒮 M ∙ ( Z ) , 𝒮 f ( M ) ∙ ( Z ′ ) ) ({\mathcal{S}}^{\bullet}_{M}(Z),{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})) f f i.e. given by M f ( S , S ′ ) = Hom 𝒮 ∙ ( Z ′ ) ( S ′ , f ( S ) ) M_{f}(S,S^{\prime})=\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(S^{\prime},f(S)) E ξ ∧ 𝒮 M ∙ ( Z ) M f → M f ∧ 𝒮 f ( M ) ∙ ( Z ′ ) E f ∘ ξ E_{\xi}\wedge_{{\mathcal{S}}^{\bullet}_{M}(Z)}M_{f}\to M_{f}\wedge_{{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})}E_{f\circ\xi}
Hom 𝒮 ∙ ( Z ) ( − , T ⊔ { ξ ( 1 ) } ) ∧ Hom 𝒮 ∙ ( Z ′ ) ( S ′ , f ( − ) ) \textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(-,T\sqcup\{\xi(1)\})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(S^{\prime},f(-))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} β ∧ α ′ ↦ f ( β ) ⋅ α ′ \scriptstyle{\beta\wedge\alpha^{\prime}\mapsto f(\beta)\cdot\alpha^{\prime}} Hom 𝒮 ∙ ( Z ′ ) ( S ′ , f ( T ) ⊔ { f ∘ ξ ( 1 ) } ) \textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(S^{\prime},f(T)\sqcup\{f\circ\xi(1)\})} Hom 𝒮 ∙ ( Z ′ ) ( − , f ( T ) ) ∧ Hom 𝒮 ∙ ( Z ′ ) ( S ′ , − ⊔ { f ∘ ξ ( 1 ) } ) \textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(-,f(T))\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(S^{\prime},-\sqcup\{f\circ\xi(1)\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ∼ \scriptstyle{\sim} β ′ ∧ α ′ ↦ ( β ′ ⊠ id f ∘ ξ ( 1 ) ) ⋅ α ′ \scriptstyle{\ \ \ \ \ \ \ \ \ \beta^{\prime}\wedge\alpha^{\prime}\mapsto(\beta^{\prime}\boxtimes\operatorname{id}\nolimits_{f\circ\xi(1)})\cdot\alpha^{\prime}}
The following lemma is a consequence of (8.1.1
0PCB
Lemma 8.1.7 . If ξ − 1 ( M ) \xi^{-1}(M)
E ξ ∧ 𝒮 M ∙ ( Z ) M f → ∼ M f ∧ 𝒮 f ( M ) ∙ ( Z ′ ) E f ∘ ξ , E_{\xi}\wedge_{{\mathcal{S}}^{\bullet}_{M}(Z)}M_{f}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}M_{f}\wedge_{{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})}E_{f\circ\xi},
and f f 2 2 L f ∘ ξ ∙ → L ξ ∙ L^{\bullet}_{f\circ\xi}\to L^{\bullet}_{\xi}
We consider now an arbitrary M M f f M f # M_{f^{\#}} ( 𝒮 f ( M ) ( Z ′ ) , 𝒮 M ( Z ) ) ({\mathcal{S}}_{f(M)}(Z^{\prime}),{\mathcal{S}}_{M}(Z)) f # f^{\#} i.e. given by
M f # ( S ′ , S ) = ⨁ p : S ′ → Z , f ∘ p = id S ′ Hom 𝒮 M ( Z ) ( S , p ( S ′ ) ) . M_{f^{\#}}(S^{\prime},S)=\bigoplus_{p:S^{\prime}\to Z,\ f\circ p=\operatorname{id}\nolimits_{S^{\prime}}}\operatorname{Hom}\nolimits_{{\mathcal{S}}_{M}(Z)}(S,p(S^{\prime})).
There is a morphism
of functors E f ∘ ξ ⊗ 𝒮 f ( M ) ( Z ′ ) M f # → M f # ⊗ 𝒮 M ( Z ) E ξ E_{f\circ\xi}\otimes_{{\mathcal{S}}_{f(M)}(Z^{\prime})}M_{f^{\#}}\to M_{f^{\#}}\otimes_{{\mathcal{S}}_{M}(Z)}E_{\xi}
⨁ p : − → Z f ∘ p = id Hom 𝒮 ( Z ′ ) ( − , T ′ ⊔ { f ∘ ξ ( 1 ) } ) ⊗ Hom 𝒮 ( Z ) ( S , p ( − ) ) \textstyle{{\displaystyle\bigoplus_{\begin{subarray}{c}p:-\to Z\\
f\circ p=\operatorname{id}\nolimits\end{subarray}}}\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z^{\prime})}(-,T^{\prime}\sqcup\{f\circ\xi(1)\})\otimes\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(S,p(-))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} β ′ ∧ α ↦ f # ( β ′ ) ⋅ α \scriptstyle{\beta^{\prime}\wedge\alpha\mapsto f^{\#}(\beta^{\prime})\cdot\alpha} ⨁ p : T ′ → Z f ∘ p = id T ′ Hom 𝒮 ( Z ) ( S , p ( T ′ ) ⊔ { ξ ( 1 ) } ) \textstyle{{\displaystyle\bigoplus_{\begin{subarray}{c}p:T^{\prime}\to Z\\
f\circ p=\operatorname{id}\nolimits_{T^{\prime}}\end{subarray}}}\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(S,p(T^{\prime})\sqcup\{\xi(1)\})} ⨁ p : T ′ → Z f ∘ p = id T ′ Hom 𝒮 ( Z ) ( − , p ( T ′ ) ) ⊗ Hom 𝒮 ( Z ) ( S , − ⊔ { ξ ( 1 ) } ) \textstyle{{\displaystyle\bigoplus_{\begin{subarray}{c}p:T^{\prime}\to Z\\
f\circ p=\operatorname{id}\nolimits_{T^{\prime}}\end{subarray}}}\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(-,p(T^{\prime}))\otimes\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(S,-\sqcup\{\xi(1)\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} ∼ \scriptstyle{\sim} β ∧ α ↦ ( β ⊠ id { ξ ( 1 ) } ) ⋅ α \scriptstyle{\ \ \ \ \ \ \ \beta\wedge\alpha\mapsto(\beta\boxtimes\operatorname{id}\nolimits_{\{\xi(1)\}})\cdot\alpha}
The following lemma is a consequence of (8.1.1
0PCC
Lemma 8.1.8 . If ξ − 1 ( M ) \xi^{-1}(M)
E f ∘ ξ ⊗ 𝒮 f ( M ) ( Z ′ ) M f # → ∼ M f # ⊗ 𝒮 M ( Z ) E ξ , E_{f\circ\xi}\otimes_{{\mathcal{S}}_{f(M)}(Z^{\prime})}M_{f^{\#}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}M_{f^{\#}}\otimes_{{\mathcal{S}}_{M}(Z)}E_{\xi},
and f # f^{\#} 2 2 L ξ → L f ∘ ξ L_{\xi}\to L_{f\circ\xi}