Approximations and adjoints in homotopy categories

We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy categories. Applications include the study of (pure) derived categories. For instance, it is shown that the pure derived category of any module category is compactly generated.

💡 Research Summary

The paper develops a general criterion for the existence of right approximations in cocomplete additive categories and shows how this criterion yields adjoint functors in homotopy categories. The work builds on a result of El Bashir, which originally dealt with locally presentable additive categories, and extends it to a much broader setting: any additive category that admits all directed colimits (i.e., is cocomplete) and contains a set of “small” objects that is sufficiently generating.

The first part of the paper revisits El Bashir’s theorem, which states that if a locally presentable additive category (\mathcal{A}) possesses a set (\mathcal{S}) of (\lambda)-presentable objects such that every object of (\mathcal{A}) is a (\lambda)-directed colimit of objects from (\mathcal{S}), then each object admits a right (\mathcal{S})-approximation. The author observes that the proof only uses the existence of directed colimits and the generating property of (\mathcal{S}); the stronger locally presentable hypothesis is unnecessary. Consequently, the paper proves the following: let (\mathcal{A}) be a cocomplete additive category and let (\mathcal{S}\subseteq\mathcal{A}) be a set of objects such that every object of (\mathcal{A}) is a directed colimit of objects from (\mathcal{S}). Then for each (X\in\mathcal{A}) there exists a morphism (f\colon S\to X) with (S\in\operatorname{Add}(\mathcal{S})) (the additive closure of (\mathcal{S})) that is right universal among morphisms from objects of (\operatorname{Add}(\mathcal{S})) to (X). In other words, a right (\mathcal{S})-approximation exists for every object. The proof constructs an “approximation tower” using directed colimits and applies Zorn’s lemma to obtain a maximal morphism satisfying the universal property.

Having established the approximation machinery, the author turns to homotopy categories. Let (\mathbf{K}(\mathcal{A})) denote the homotopy category of chain complexes over (\mathcal{A}). By taking the set (\mathcal{S}) of small objects in (\mathcal{A}) and forming its additive closure under shifts and finite direct sums, one obtains a set (\mathcal{S}^{\mathrm{cpx}}) of compact complexes inside (\mathbf{K}(\mathcal{A})). The right approximation result applied objectwise to complexes shows that every complex admits a right (\mathcal{S}^{\mathrm{cpx}})-approximation. Consequently, the inclusion functor \