Splitting lemma for stable ∞-categories

Exercise 4.2.18 (Splitting lemma). Let \(X \xrightarrow {i} Y \xrightarrow {p} Z\) be an exact sequence in a stable \(\infty \)-category \(C\). The following are equivalent:

(1)

The map \(i\) admits a retraction \(r\colon Y \to X\) (i.e., \(ri \simeq \id _X\));

(2)

The map \(p\) admits a section \(s\colon Z \to Y\) (i.e., \(ps \simeq \id _Z\));

(3)

There is an isomorphism \(Y \cong X \oplus Z\) making the following diagram commute:

Hint. Given a retraction of \(i\), paste the two resulting pushout squares to identify \(Y\) with \(X\oplus Z\). The argument starting from a section of \(p\) is dual.