Derived ∞-categories are stable

Proof. Assuming stability, the last claim is immediate from part (4) of the previous corollary. For stability, we saw in Corollary 6.1.29 that \(\D (\Aa )\) admits finite limits and finite colimits. Moreover, the localization functor \(\gamma \colon \Ch (\Aa ) \to \D (\Aa )\) preserves the zero object, so \(\D (\Aa )\) is pointed. By Theorem 4.2.2, it remains to show that the unit and counit of the adjunction \(\Sigma \dashv \Omega \) is an equivalence. By the universal property of \(\gamma \colon \Ch (\Aa ) \to \D (\Aa )\), this amounts to showing that for a chain complex \(C_{\bullet }\), the unit \(\gamma (C_{\bullet }) \to \Omega \Sigma \gamma (C_{\bullet })\) and the counit \(\Sigma \Omega \gamma (C_{\bullet }) \to \gamma (C_{\bullet })\) are isomorphisms in \(\D (\Aa )\).

To this end, consider the chain complex \(C'_{\bullet }\) defined by \(C'_n := C_n \oplus C_{n-1}\), with differential \(d^{C'}_n(c_n,c_{n-1}) = (d^Cc_n + c_{n-1}, -d^Cc_{n-1})\); note that this is the mapping cylinder of the map \(C_{\bullet } \to 0\). The maps \(f(c_n) := (c_n,0)\) and \(g(c_n,c_{n-1}) := c_{n-1}\) define a short exact sequence \[ 0 \to C_{\bullet } \xhookrightarrow {f} C'_{\bullet } \overset {g}{\twoheadrightarrow } C[1]_{\bullet } \to 0, \] and by part (4) of Corollary 6.1.29 the resulting commutative square

in \(\D (\Aa )\) is both a pullback square and a pushout square. Since the map \(C'_{\bullet } \to 0\) is a quasi-isomorphism, we get \(\gamma (C'_{\bullet }) \simeq 0\), so this square simultaneously exhibits \(\gamma (C[1]_{\bullet })\) as the suspension of \(\gamma (C_{\bullet })\) and \(\gamma (C_{\bullet })\) as the loops of \(\gamma (C[1]_{\bullet })\). We conclude that the maps \(\gamma (C_{\bullet }) \to \Omega \Sigma \gamma (C_{\bullet })\) and \(\Sigma \Omega \gamma (C_{\bullet }) \to \gamma (C_{\bullet })\) are isomorphisms in \(\D (\Aa )\). □