Corollary 4.4.35 (Dual mapping telescope). Let \(C\) be a stable \(\infty \)-category with countable products. For every tower \[ \dots \xrightarrow {f_2} Y_2 \xrightarrow {f_1} Y_1 \xrightarrow {f_0} Y_0 \] in \(C\), there is an exact sequence \[ \lim _k Y_k \longrightarrow \prod _{k \geq 0}Y_k \xrightarrow {\ \id -f\ } \prod _{k \geq 0}Y_k, \] where the \(k\)-th component of \((\id -f)(y)\) is \(y_k-f_k(y_{k+1})\).
Proof. By Lemma 2.4.29, the colimit \(\colim _kY_k\) is the pushout of the two maps from \(\bigoplus _kY_k\) to the coproducts of the even and odd terms. In a stable \(\infty \)-category, the pushout of a span \(B \xleftarrow {a} A \xrightarrow {b} C\) is the cofiber of \[ (a,-b)\colon A \longrightarrow B \oplus C. \] After identifying the coproduct of the even and odd terms with \(\bigoplus _k Y_k\) and changing the signs on the odd source summands, this map is precisely \(\id -f\). □
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