Lemma 4.4.34 (Mapping telescope). Let \(C\) be a stable \(\infty \)-category with countable coproducts, and let \(Y_0 \xrightarrow {f_0} Y_1 \xrightarrow {f_1} Y_2 \to \dots \) be a sequential diagram in \(C\). Write \(\iota _k\colon Y_k \to \bigoplus _{j \geq 0} Y_j\) for the inclusion of the \(k\)-th summand. Then there is an exact sequence \[ \bigoplus _{k \geq 0} Y_k \xrightarrow {\ \id - f\ } \bigoplus _{k \geq 0} Y_k \longrightarrow \colim _k Y_k, \] where \(\id - f\) restricts on the \(k\)-th summand to the difference \(\iota _k - \iota _{k+1} \circ f_k\).
Proof. The natural isomorphism \(\Sigma ^{\infty }(\Sigma X)\cong (\Sigma ^{\infty }X)[1]\) gives suspension isomorphisms \[ \widetilde E_{k+1}(\Sigma X)=\pi _{k+1}((\Sigma ^{\infty }X\otimes E)[1])\cong \pi _k(\Sigma ^{\infty }X\otimes E)=\widetilde E_k(X). \] Both \(\Sigma ^{\infty }\) and \(-\otimes E\) preserve cofiber sequences. Exactness therefore follows from the long exact sequence of homotopy groups in Proposition 4.4.30.
It remains to verify the wedge axiom. The functor \(\Sigma ^{\infty }(-)\otimes E\) preserves coproducts, and each \(\pi _k\colon \Sp \to \Ab \) preserves coproducts by Lemma 4.4.29. Thus \[ \bigoplus _{i\in I}\widetilde E_k(X_i)\xrightarrow {\cong }\widetilde E_k\left (\bigvee _{i\in I}X_i\right ), \] as required. □
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