Reduced free-chain adjunction for local derived Eilenberg targets

The category of objects $\ast \to S$ inherits the local simplicial model: underlying factorizations acquire their basepoints by composition, and the lifts in pointed lifting squares preserve the specified basepoints. Every pointed $S$ is cofibrant because $\ast \to S$ is a monomorphism. Additive maps from the quotient ${\tilde{\mathbb{Z}}}_{\mathrm{s}\mathrm{h}}[S]$ to $A$ are precisely pointed maps $S\to uA$. The right adjoint $u_{\ast }$ preserves fibrations and trivial fibrations by [F1], so this reduced-free/pointed-forgetful adjunction is Quillen. It is simplicial: the same quotient universal property identifies maps out of the pointed tensor $S\wedge \mathrm{\Delta }[q{]}_{+}$, the sheaf quotient of the sheafified product $S\times \mathrm{\Delta }[q]$ with its basepoint copy collapsed, with additive maps out of ${\tilde{\mathbb{Z}}}_{\mathrm{s}\mathrm{h}}[S]\otimes \mathbb{Z}[\mathrm{\Delta }[q]]$, in every simplicial degree. The augmentation sends every free generator to $1$ and is split by the given basepoint. Its kernel identifies with the quotient by that basepoint via the explicit mutually inverse maps $[s]<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>[\ast ]\mapsto [s]-[\ast ]$ and the quotient projection restricted to the kernel. Thus the unreduced simplicial abelian sheaf splits naturally as the basepoint summand plus the reduced one. Normalization and exact sheafification preserve this splitting and compute the stated local homology sheaves; a pointed chain map is identity on the basepoint summand. An unreduced quasi-isomorphism therefore induces a reduced quasi-isomorphism.

Set $E={\tau }^{\le 0}(D[n])$ and take $E\to I$ as stated. The good-truncation result [F2] ensures that changing $D$ by a quasi-isomorphism changes $E$ by one. By [F1], $\mathrm{\Gamma }I$ is fibrant in the additive model and $u\mathrm{\Gamma }I$ is fibrant in the pointed local model. Step 1.1 makes ${\tilde{\mathbb{Z}}}_{\mathrm{s}\mathrm{h}}[S]$ cofibrant, hence its normalization is cofibrant in the transported model. Applying the strict simplicial adjunction to these cofibrant and fibrant representatives, and then the normalization equivalence, identifies their actual function complexes. They compute exactly the two displayed derived mapping spaces. Replacement invariance in [F1] gives independence of $I$, naturality in the localized coefficient and source objects, and invariance under quasi-isomorphisms of reduced chains. This proves the full mapping-space assertion, rather than only a bijection on components.