Stable Yoneda Waldhausen presentations without retract completion

Form $P_D$ using [F1] and let $C_D$ consist of all its cofibrant objects whose derived presheaves are stably representable. The spectral Yoneda embedding preserves finite limits: evaluation at each $d$ gives the mapping-spectrum functor, which preserves limits in its second variable. Since both $D$ and spectral presheaves are stable, finite limits and finite colimits coincide, so this embedding is exact. Its full replete image is therefore closed under zero and finite homotopy pushouts, without adding retracts. The model zero is cofibrant and belongs to $C_D$. For a map $X\to Y$ in $C_D$, model factorization $X\hookrightarrow Z\stackrel{\sim }{\to }Y$ has cofibrant $Z$, and $Z$ is still stably representable because it is weakly equivalent to $Y$. For a span $Y\leftarrow X\to Z$ of cofibrant objects with $X\to Y$ a cofibration, its model pushout $T$ is cofibrant: $Z\to T$ is a cofibration. For every fibrant test object $B$, the enriched adjunction identifies $\mathrm{map}(T,B)$ with $\mathrm{map}(Y,B){\times }_{\mathrm{map}(X,B)}\mathrm{map}(Z,B)$. The restriction from $Y$ is a fibration by [F2], so this is a homotopy pullback. Thus $T$ computes the ambient homotopy pushout and belongs to $C_D$ by exactness of Yoneda. The same mapping-space calculation shows that objectwise weak equivalences between these spans induce weak equivalences on pushouts, giving gluing. Closure of cofibrations under composition and pushout follows from their lifting characterization in [A1]; weak equivalences are closed under composition by two-out-of-three. These facts verify every stated Waldhausen axiom and factorization.

The infinity-localization of $C_D$ is the full subcategory modeled by its cofibrant-fibrant objects. To justify this restriction rather than assume it, regard $C_D$ as a full simplicial subcategory of the cofibrants of $P_D$. It is closed under $\mathrm{\Delta }[1]\otimes X$ by [F2] and its defining weak-equivalence closure. Thus the four hypotheses of Proposition 1.3.4.7 hold: isomorphisms and two-out-of-three are inherited, interval objects satisfy the enriched adjunction, and interval projection is a weak equivalence. Every $X$ also has a trivial-cofibration fibrant replacement $X\to RX$ remaining in $C_D$. Into any cofibrant-fibrant $B$, this replacement induces an equivalence of mapping spaces by the corner axiom. Consequently the fibrant-replacement adjunction and marked-localization argument in the proof of Theorem 1.3.4.20 restrict to this full simplicial subcategory: their units are these weak equivalences and their mapping spaces into the fibrant objects are unchanged. They identify $N(C_D)[W_D^{-1}]$ with the nerve of the full simplicial category of its cofibrant-fibrant objects. That category is exactly ${\mathrm{\Omega }}^{\mathrm{\infty }}\mathrm{{\rm Y}}(D)$, so [F1] identifies the localization fully with $D$. In particular it embeds fully in the underlying infinity-category of $P_D$; a map inverted in it is an equivalence of spectral presheaves, hence an objectwise equivalence of spectra, and therefore a projective model weak equivalence by [F1]. The converse holds by the defining localization, proving DHKS-saturation. Step 1.1 and this full embedding identify its homotopy pushouts with those in $D$. Finally all $V$-small presheaf structures and morphisms on the fixed small indexing category form sets in $W$, by the universe closure convention; the full subcategory used here is therefore $W$-small. The model operations are performed in $V$ and their results are retained in this full category, not discarded by taking an ordinary skeleton. Larger compatible universes are assumed if another category of these data is needed. Choice enters the imported realizations and replacements through [A1]; no functorial strict selection of $C_D$ as $D$ varies is claimed.