Definition 6.6.8 (Projective chain complex). Let \(\Aa \) be an abelian category. We write \(\Ch ^-(\Aa )\subseteq \Ch (\Aa )\) for the full subcategory of bounded below chain complexes, those \(C_{\bullet }\) for which \(C_n=0\) for all sufficiently small \(n\). A chain complex \(P_{\bullet } \in \Ch (\Aa )\) is called projective if each object \(P_n\) is projective in \(\Aa \), in the sense of Section 6.4.1 We write \(\Ch ^-_{\proj }(\Aa ) \subseteq \Ch ^-(\Aa )\) for the full subcategory of bounded below projective chain complexes.
Notes
1This is weaker than \(P_{\bullet }\) being a projective object of the abelian category \(\Ch (\Aa )\): the latter are exactly the contractible complexes that are degreewise projective. Throughout, ‘projective chain complex’ always refers to the degreewise condition; some authors instead say degreewise projective or complex of projectives.
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