External dg-nerve mapping and complex-localization bridges

PDF p. 16: Perfect complexes with support

Assume Choice and the fixed universes of the derived enhancement. For module complexes, use the differential graded nerve $N_{\mathrm{d}\mathrm{g}}$ of the category enriched in Hom complexes. Quasi-isomorphisms of Hom complexes induce equivalences of the associated mapping spaces. Its reflective derived localization $L$ has unit a quasi-isomorphism $Q\to I_Q$ to a fibrant complex, and adjunction identifies ${\mathrm{Map}}_{\mathcal{D}(A)}(LP,LQ)$ with ${\mathrm{Map}}_{N_{\mathrm{d}\mathrm{g}}\mathrm{C}\mathrm{h}(A)}(P,I_Q)$, naturally in $P,Q$. The composite from ordinary complexes is the quasi-isomorphism localization defining the supplied $\mathcal{D}(A)$.

For a full dg subcategory $\mathcal{C}$ of complexes in an additive category, closed under identity mapping cylinders, its ordinary category localized at cochain homotopy equivalences is equivalent to $N_{\mathrm{d}\mathrm{g}}\mathcal{C}$ by the canonical inclusion.

External input on derived enhancements

The Hom complex of chain complexes

The mapping cone of a chain map

The Axiom of Choice

Grothendieck universe closure convention

Given: Module complexes and, in the last assertion, a full dg subcategory with the stated cylinder closure.

Goal: Import the mapping-space and localization bridges, not a proof of perfect-model equivalence.

[A1] The Axiom of Choice and Grothendieck universe closure convention retain the assumptions of External input on derived enhancements, including the module categories and their unbounded derived localization. Universes containing the data and $\omega$ are assumed to exist; this is not deduced from Choice.

[F1] The Hom complex of chain complexes supplies the Hom enrichment. Read its chain grading as $C_n^{\mathrm{♯}}=C^{-n}$; its Hom degree $r$ is cochain degree $-r$. Composition is composition of graded maps, with its usual graded Leibniz identity. The dg nerve is the construction imported from reference 0, rather than the nerve of the ordinary category obtained by forgetting higher homotopies.

[F2] Identity mapping cylinders are those of reference 0, equivalently $P\oplus \mathrm{Cone}({\mathrm{i}\mathrm{d}}_P)$ with the usual endpoint maps. The mapping cone of a chain map supplies cones after the same reversal of grading. A cochain homotopy equivalence is a cochain map admitting an inverse up to homotopies $f-g=dh+hd$, as in the supplied derived enhancement.

Import reference 0 under [A1], using the cochain translation [F1] and the cylinder convention [F2]. The adjunction and nerve comparisons are established external inputs; no local proof is claimed.