Chain complexes

Example 20.1.7 (Chain complexes). Let \(\Aa \) be a symmetric monoidal abelian category with enough projectives. Assume that its monoidal unit is projective and that the tensor product of two projective objects is again projective. Equip the category \(\Ch ^{-}(\Aa )\) of bounded below chain complexes with the projective cofibration structure from Proposition 6.6.15 and the symmetric monoidal structure from Definition 6.6.23. We claim that this monoidal structure is left derivable. To see this, consider finitely many bounded below projective chain complexes \(C^1_{\bullet }, \dots , C^n_{\bullet }\), finitely many tensor-cofibrant bounded below chain complexes \(D^1_{\bullet }, \dots , D^n_{\bullet }\), and quasi-isomorphisms \(f_i\colon C^i_{\bullet } \to D^i_{\bullet }\). By assumption on \(\Aa \), each \(D^i_{\bullet }\) is again a bounded below projective chain complex. Hence by Proposition 6.6.11, each \(f_i\) is actually a chain homotopy equivalence. Since the tensor product of chain maps preserves chain homotopies in each variable, it follows by induction on \(n\) that the tensor product \[ \bigotimes _{i=1}^n f_i\colon \bigotimes _{i=1}^n C^i_{\bullet } \to \bigotimes _{i=1}^n D^i_{\bullet } \] is again a chain homotopy equivalence, and hence a quasi-isomorphism.

Thus Theorem 20.1.6 equips \(\D ^-(\Aa )\) with a symmetric monoidal structure and the localization \(\Ch ^-(\Aa )\to \D ^-(\Aa )\) with a lax symmetric monoidal refinement. For comparison with the construction in Chapter 6, Proposition 6.6.16 identifies \(\D ^-(\Aa )\) with the localization of the symmetric monoidal category \(\Ch ^-_{\proj }(\Aa )\) at the quasi-isomorphisms. Since tensor product preserves quasi-isomorphisms between projective complexes in each variable, Proposition 20.1.1 gives a second description of the induced tensor product, namely the derived tensor product constructed in Definition 6.6.25.