Limits and colimits in derived ∞-categories

Corollary 6.1.29. Let \(\Aa \) be an abelian category.

(1)

The derived \(\infty \)-category \(\D (\Aa )\) admits finite limits and finite colimits.

(2)

If \(\Aa \) satisfies (AB3) and (AB4), then \(\D (\Aa )\) admits small colimits. If it satisfies (AB3\(^*\)) and (AB4\(^*\)) then \(\D (\Aa )\) admits small limits.

(3)

The localization functor \(\gamma \colon \Ch (\Aa ) \to \D (\Aa )\) preserves pushouts along monomorphisms and preserves pullbacks along epimorphisms.

(4)

For a short exact sequence \(0 \to C_{\bullet } \hookrightarrow D_{\bullet } \twoheadrightarrow E_{\bullet } \to 0\) in \(\Ch (\Aa )\), the induced sequence \[ \gamma (C_{\bullet }) \to \gamma (D_{\bullet }) \to \gamma (E_{\bullet }) \] in \(\D (\Aa )\) is both a fiber sequence and a cofiber sequence.