0NPF
Theorem 9.10 . Let 𝒜 {\mathcal{A}} ∞ \infty
Map ( 𝒰 wloc κ ¯ ( 𝒮 ∞ ω ) , 𝒰 wloc κ ¯ ( 𝒜 ) ) ≃ K ( 𝒜 ) . \mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq K({\mathcal{A}})\,.
0NPG
Proof. By construction, the object 𝒰 add κ ¯ ( 𝒮 ∞ ω ) \underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}) ℳ add κ ¯ \underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}} S S 8.4 S ¯ \overline{S} S S [52 , 5.5.4.5] , and let X X S S 𝒰 add κ ¯ ( 𝒜 ) → X \underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X S S 𝒰 add κ ¯ ( 𝒜 ) → X \underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X S ¯ \overline{S}
Map ( 𝒰 wloc κ ¯ ( 𝒮 ∞ ω ) , 𝒰 wloc κ ¯ ( 𝒜 ) ) ≃ Map ( 𝒰 add κ ¯ ( 𝒮 ∞ ω ) , X ) , \mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),X),
so it suffices to show that the functor
(9.11) R := Map ( 𝒰 add κ ¯ ( 𝒮 ∞ ω ) , − ) : ℳ add κ ¯ ⟶ 𝒮 ∞ R:=\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),-):\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\longrightarrow{\mathcal{S}}_{\infty}
sends the maps in S ¯ \overline{S} ℳ add κ ¯ \underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}} ∞ \infty 𝒰 add κ ¯ ( 𝒮 ∞ ω ) \underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}) R R R R S S
Consider the following diagram
(9.12) 𝒰 add κ ¯ ( 𝒜 ) \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒰 add κ ¯ ( ℬ ) \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒰 add κ ¯ ( ℬ ) / 𝒰 add κ ¯ ( 𝒜 ) \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})/\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒰 add κ ¯ ( 𝒜 ) \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒰 add κ ¯ ( ℬ ) \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} 𝒰 add κ ¯ ( ℬ / 𝒜 ) . \textstyle{\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}}/{\mathcal{A}})\,.}
By applying the functor (9.11 9.12 9.9 𝒮 \mathcal{S}
(9.13) K ( 𝒜 ) \textstyle{K({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} K ( ℬ ) \textstyle{K({\mathcal{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} K ( ℬ ) / K ( 𝒜 ) \textstyle{K({\mathcal{B}})/K({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} K ( 𝒜 ) \textstyle{K({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} K ( ℬ ) \textstyle{K({\mathcal{B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} K ( ℬ / 𝒜 ) , \textstyle{K({\mathcal{B}}/{\mathcal{A}})\,,}
where the upper row is a homotopy cofiber sequence. Now, an
argument analogous to the one used in the proof of
proposition 7.19 9.13