Replacement-invariant derived enriched mapping spaces

Definition and fibrancy. For objects $X,Y$, let $X_c\to X$ be the functorial cofibrant replacement and $Y\to Y_f$ the functorial fibrant replacement; define $\mathrm{R}\mathrm{M}\mathrm{a}\mathrm{p}(X,Y)=\mathrm{M}\mathrm{a}\mathrm{p}(X_c,Y_f)$. The mapping corner axiom of [F2] shows that $\mathrm{R}\mathrm{M}\mathrm{a}\mathrm{p}(X,Y)$ is a Kan simplicial set, since the source $X_c$ is cofibrant and the target $Y_f$ is fibrant.

Target invariance. Let $C$ be cofibrant and let $D\to D^{\mathrm{\prime }}$ be a weak equivalence between fibrant objects. Factor it as a trivial cofibration $D\to E$ followed by a trivial fibration $E\to D^{\mathrm{\prime }}$, with $E$ fibrant. A trivial cofibration between fibrant objects is a homotopy equivalence: lifting against the terminal fibration $D\to {\ast }_{\mathcal{M}}$ gives a retraction $r\text{\hspace{0.17em}⁣}:E\to D$ with $ri={\mathrm{i}\mathrm{d}}_D$ (in a slice over $B$, ${\ast }_{\mathcal{M}}$ is $B\stackrel{\mathrm{i}\mathrm{d}}{\to }B$ and the lifting square is a square over $B$). Lifting against the endpoint fibration $\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}(E)\to E{\times }_{{\ast }_{\mathcal{M}}}E$, using the constant path on $i$ as the map from $D$ and $(ir,{\mathrm{i}\mathrm{d}}_E)$ as the map from $E$, gives a homotopy $ir\simeq {\mathrm{i}\mathrm{d}}_E$; in a slice the path object is $E^{\mathrm{\Delta }[1]}{\times }_{B^{\mathrm{\Delta }[1]}}B$ and both endpoints lie over the same section of $B$. Mapping from the cofibrant $C$ preserves simplicial homotopies and sends trivial fibrations to boundary-trivial maps by the corner axiom, which are homotopy equivalences by [F3]. Hence $\mathrm{M}\mathrm{a}\mathrm{p}(C,D)\to \mathrm{M}\mathrm{a}\mathrm{p}(C,E)\to \mathrm{M}\mathrm{a}\mathrm{p}(C,D^{\mathrm{\prime }})$ are homotopy equivalences, proving invariance in the target.

Source invariance. Let $D$ be fibrant and let $C\to C^{\mathrm{\prime }}$ be a weak equivalence between cofibrant objects. A trivial cofibration between cofibrant objects becomes boundary-trivial after mapping into $D$ by the corner axiom; a trivial fibration $q\text{\hspace{0.17em}⁣}:C\to C^{\mathrm{\prime }}$ between cofibrant objects has a section $s$ obtained by lifting $\mathrm{\varnothing }\to C^{\mathrm{\prime }}$ through $q$ (using cofibrancy of $C^{\mathrm{\prime }}$), and the cotensor corner $q^{\mathrm{\partial }\mathrm{\Delta }[1]}$ is boundary-trivial by [F2]; lifting the cofibration $\mathrm{\varnothing }\to C$ into it with endpoints $sq$ and ${\mathrm{i}\mathrm{d}}_C$ and the constant path on $q$ produces a homotopy $sq\simeq {\mathrm{i}\mathrm{d}}_C$ over $C^{\mathrm{\prime }}$, so $q$ is a homotopy equivalence and the contravariant mapping maps are homotopy equivalences. Factoring a general weak equivalence between cofibrant objects into a trivial cofibration followed by a trivial fibration gives invariance in the source.

Enriched adjunction. Let $L⊣R$ be an enriched Quillen adjunction, so $R$ preserves fibrations and trivial fibrations. For cofibrant $C$ and fibrant $D$ the enriched adjunction gives a strict isomorphism $\mathrm{M}\mathrm{a}\mathrm{p}(LC,D)\cong \mathrm{M}\mathrm{a}\mathrm{p}(C,RD)$, and $LC$ is cofibrant while $RD$ is fibrant; composing with the replacement comparisons of steps 2.1 and 2.2 yields the canonical derived mapping equivalence $\mathrm{R}\mathrm{M}\mathrm{a}\mathrm{p}(LX,Y)\simeq \mathrm{R}\mathrm{M}\mathrm{a}\mathrm{p}(X,RY)$ for cofibrant-fibrant representatives. The Quillen adjunction condition itself is the lifting formulation of Model categories and Quillen adjunctions.

The ${\pi }_0$ interface. The functorial cofibrant then fibrant replacement supplies natural weak-equivalence zigzags between every object and a cofibrant-fibrant model, and weak maps between such models are homotopy equivalences by steps 2.1-2.2. Simplicially homotopic maps agree in the localization because the constant-path map is a weak equivalence with both endpoints as inverses; hence the category of homotopy classes of maps between cofibrant-fibrant models has the universal localization property for the weak equivalences. This proves the asserted ${\pi }_0$ description with no independent hammock, infinity-categorical localization or coherent-diagram strictification claimed.