For an abelian category $\mathcal{ A}$ we investigate when the stable categories $\underline{ \mathrm{ GPro}}\mathrm{ j}(\mathcal{ A})$ and $\underline{ \mathrm{ GIn}}\mathrm{ j}(\mathcal{ A})$ are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on $\mathcal{ A}$ that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from $\mathcal{ A}$ to categories naturally associated to $\mathcal{ A}$, such as $\mathrm{ Ch}(\mathcal{ A})$, the category of chain complexes in $\mathcal{ A}$, or $\mathrm{ Rep}(Q,\mathcal{ A})$, the category of $\mathcal{ A}$-valued representations of a quiver $Q$.
Categories: math.CT math.AC math.AT math.RT