Cofibration structure on chain complexes

Proof. We verify the (dualized) axioms from Definition 2.2.4:

(1) The 2-out-of-3 property for quasi-isomorphisms follows immediately from the 2-out-of-3 property for isomorphisms in \(\Aa \).

(2) All isomorphisms are monomorphisms, and since \(\Ch (\Aa )\) is again an abelian category the monomorphisms are closed under compositions and pushouts; see Proposition 6.1.5. Note that every chain complex is cofibrant, since the map \(0 \to C\) is a monomorphism for all \(C \in \Ch (\Aa )\).

(3) Consider a pushout square in \(\Ch (\Aa )\) of the form

where \(i\) and \(i'\) are monomorphisms and \(i\) is a quasi-isomorphism. We need to show that also \(i'\) is a quasi-isomorphism. Let \(E_{\bullet }\) be the cokernel of \(i\) and let \(E'_{\bullet }\) be the cokernel of \(i'\). Since \(i\) is a quasi-isomorphism, it follows from Corollary 6.1.18 that the homology of \(E_{\bullet }\) is trivial. By the pasting law for pushout squares, we see that the induced map \(E_{\bullet } \to E'_{\bullet }\) is an isomorphism of chain complexes, and so also the homology of \(E'_{\bullet }\) is trivial. By applying Corollary 6.1.18 in the other direction, it follows that \(i'\) is a quasi-isomorphism.

(4) We now show that any chain map \(f\colon C_{\bullet } \to D_{\bullet }\) factors functorially as a monomorphism followed by a quasi-isomorphism. Define the mapping cylinder \(\Cyl (f)\) of \(f\) as the chain complex defined by \((\Cyl (f))_n = C_n \oplus C_{n-1} \oplus D_n\). The differential \(d^{\Cyl (f)}\colon (\Cyl (f))_n \to (\Cyl (f))_{n-1}\) is given by \[ d^{\Cyl (f)} \quad := \quad \begin {pmatrix} d^C & \id _C & 0 \\ 0 & -d^C & 0 \\ 0 & -f & d^D \end {pmatrix}, \] i.e. in terms of elements we have \(d^{\Cyl (f)}(c_n, c_{n-1}, d_n) := (d^C c_n + c_{n-1}, -d^C c_{n-1}, -f(c_{n-1}) + d^D d_n)\). There is a chain map \(i_C\colon C_{\bullet } \to \Cyl (f)_{\bullet }\) given by \(i_C(c_n) = (c_n, 0, 0)\), and a chain map \(p\colon \Cyl (f)_{\bullet } \to D_{\bullet }\) given by \(p(c_n, c_{n-1}, d_n) = f(c_{n}) + d_n\). Note that the constructions of \(\Cyl (f)\), \(i_C\) and \(p\) are functorial in \(f\) and that \(f = p \circ i_C\), so this provides a functorial factorization of \(f\). Since \(i_C\) is clearly a monomorphism, it remains to show that \(p\) is a quasi-isomorphism. By Exercise 6.1.16, it suffices to show that \(p\) is in fact a chain homotopy equivalence. We claim that the chain map \(i_D\colon D_{\bullet } \to \Cyl (f)_{\bullet }\) given by \(i_D(d_n) = (0, 0, d_n)\) is a chain homotopy inverse for \(p\). It is clear that \(p \circ i_D = \id _{D_{\bullet }}\), so it remains to construct a chain homotopy \(H\) between \(i_D \circ p\) and \(\id _{\Cyl (f)_{\bullet }}\). Such a chain homotopy is given by \(H_n(c_n,c_{n-1},d_n) := (0, c_n,0)\).

We have thus shown that the triple \((\Ch (\Aa ), \textup {quasi-isos}, \textup {monos})\) is an \(\infty \)-category with weak equivalences and cofibrations, and that it has functorial factorizations. Under the assumption that \(\Aa \) satisfies (AB3) and (AB4), the category \(\Ch (\Aa )\) inherits arbitrary coproducts from \(\Aa \), formed degreewise, and monomorphisms in \(\Ch (\Aa )\) are again closed under arbitrary coproducts. Moreover, since the coproduct functor \(\bigoplus _I \colon \Ch (\Aa )^I \to \Ch (\Aa )\) is exact, it commutes with formation of homology, and it follows that an arbitrary coproduct of quasi-isomorphisms is again a quasi-isomorphism. This verifies the (dualized) conditions from Definition 2.2.10, showing that \(\Ch (\Aa )\) is homotopy cocomplete. □