Derived hom in the bounded setting

Let $M\in D^{-}(\mathcal{A})$ and $N\in D^{+}(\mathcal{A})$, with termwise bounded representatives. With supplied bounded-above projective models under Projective complexes model the bounded above derived category, define $\mathbf{R}\mathrm{Hom}(M,N)=Q\underset{}{\mathrm{Hom}}(P_M,N)$. Alternatively, with supplied bounded-below injective models under Injective complexes model the bounded below derived category, use $Q\underset{}{\mathrm{Hom}}(M,I_N)$. The target is $D^{+}(\mathbf{A}\mathbf{b})$.

Here the cochain form of The Hom complex of chain complexes has degree-$r$ term ${\prod }_i\mathrm{Hom}(M^i,N^{i+r})$ and differential $du=d_{N}u-(-1{)}^{r}u{d}_M$. If $M^i=0$ above $b$ and $N^j=0$ below $a$, nonzero factors require $a-r\le i\le b$, a finite interval, and the whole term is zero for $r<a-b$.

This construction is a bifunctor on the declared derived categories. Indeed homotopies in either variable induce Hom-complex homotopies. Replacing a projective model by a homotopy equivalent one therefore changes its Hom complex by a homotopy equivalence. A quasi-isomorphism in the target has acyclic cone, whose Hom from $P_M$ is acyclic by K-projectivity; hence the Hom map is a quasi-isomorphism. This also follows degree by degree from Morphisms from a homotopically projective complex need no roof, which identifies each Hom-complex cohomology with the corresponding derived Hom. The injective argument uses Morphisms into a homotopically injective complex need no roof and reverses the roles of source and target. Thus both variables descend through localization. When both systems exist, the quasi-isomorphisms $\underset{}{\mathrm{Hom}}(P_M,N)\to \underset{}{\mathrm{Hom}}(P_M,I_N)\leftarrow \underset{}{\mathrm{Hom}}(M,I_N)$ give their natural identification. Either one-sided resolution hypothesis suffices.