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A model-categorical enhancement of D(k-filt) is constructed in arXiv:1602.01515, see Corollary 3.58 there. It also shows that this model category is combinatorial and compactly generated, as requested. A morphism f: k→k' of Z-filtered rings induces a Quillen adjunction, as shown in Theorem 5.5 there. In fact, if f is a graded equivalence, then the Quillen ...


I don't think this question is really meaningful at the level of 1-categories, but assuming that what your after is the model category theoretic version of reflecting into a reflexive $\infty$-subcategory, then what your looking for is the fibrant replacements arising from (left) Bousfield localizations of simplicial model categories. More precisely, suppose ...


The generating cofibrations are the inclusions $\emptyset \hookrightarrow \{ \ast \}$ and $\{ 0, 1 \} \hookrightarrow \{ 0 \to 1 \}$ plus the projection $\{ 0 \rightrightarrows 1 \} \to \{ 0 \rightarrow 1 \}$. It is easy to see that a functor has the right lifting property with respect to these if and only if it is fully faithful and surjective on objects.

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