On the Dreaded Right Bousfield Localization

I verify the existence of right Bousfield localizations of right semimodel categories, and I apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right semimodel category.

💡 Research Summary

The paper addresses a long‑standing gap in homotopical algebra: the existence of right Bousfield localizations in contexts where a full model structure is unavailable or unsuitable. Classical Bousfield localization theory is well‑developed for left localizations, which rely on left properness and the existence of cofibrant replacements. Right localizations, by contrast, require right properness and are notoriously difficult to construct when the underlying category lacks sufficient fibrancy or properness. The author circumvents these obstacles by working within the more flexible framework of right semimodel categories. A right semimodel category retains the usual cofibration, weak equivalence, and fibrant‑object structure, but the lifting axioms for trivial cofibrations are required only against fibrant objects, not against all objects. This relaxation makes it possible to define and manipulate homotopical structures even when right properness fails.

The first major contribution is a general existence theorem for right Bousfield localizations of right semimodel categories. Starting with a right semimodel category ( \mathcal{C} ) and a set ( S ) of maps that one wishes to invert, the author adapts Smith’s small‑object argument to the semimodel setting. The key technical steps are: (1) showing that the class of ( S )-local objects can be generated by a set of small objects, (2) constructing a new class of weak equivalences—those maps that become isomorphisms after applying the derived hom‑functor into every ( S )-local object—and (3) proving that the resulting three‑tuple (cofibrations, new weak equivalences, fibrations) satisfies the axioms of a right semimodel category. Crucially, the proof does not require the ambient category to be right proper; the semimodel axioms are sufficient to control the necessary lifting properties. This theorem thus extends the classical right‑localization existence results to a far broader class of homotopical settings.

In the second part, the author applies the existence theorem to the problem of modeling homotopy limits of left Quillen presheaves. A left Quillen presheaf consists of a diagram ( { \mathcal{M}i }{i\in I} ) of model categories together with left Quillen functors along the morphisms of a small indexing category ( I ). The homotopy limit of such a diagram is traditionally obtained by endowing the category of sections ( \mathrm{Sect}(\mathcal{M}_\bullet) ) with a projective model structure and then performing a left Bousfield localization that forces the section to be homotopy‑compatible with the transition maps. However, when the individual ( \mathcal{M}_i ) are not left proper, or when the projective model structure fails to exist as a full model category, this approach breaks down.

The paper resolves this by first constructing a right semimodel projective structure on the section category. Here, cofibrations are defined objectwise, fibrations are those maps that are objectwise fibrations between fibrant objects, and weak equivalences are objectwise weak equivalences. Because the lifting axioms are required only against fibrant sections, the construction works even when some ( \mathcal{M}i ) lack left properness. Next, the author identifies a set ( S ) of maps encoding the compatibility conditions that a section must satisfy to be a homotopy limit (essentially the derived unit maps of the left Quillen functors). Applying the right Bousfield localization theorem proved earlier, the author localizes the right semimodel projective structure at ( S ). The resulting localized right semimodel category ( L_S\mathrm{Sect}(\mathcal{M}\bullet) ) has the following crucial properties:

1. Fibrant objects are precisely the homotopy‑limit sections: objectwise fibrant sections equipped with homotopy‑coherent comparison maps that become weak equivalences after applying the derived left Quillen functors.
2. Homotopy classes of maps between fibrant objects compute the derived mapping spaces in the homotopy limit of the original diagram.
3. The construction is functorial in the presheaf, allowing one to transport model‑categorical structures along natural transformations of left Quillen presheaves.

The author demonstrates the utility of this framework through several examples. One example treats a diagram of chain complexes over varying rings where the change‑of‑ring functors are left Quillen but not left proper; the localized right semimodel category correctly models the derived inverse limit of the diagram. Another example involves a diagram of spectral model categories indexed by a simplicial category, illustrating how the method accommodates higher‑categorical indexing.

Finally, the paper discusses broader implications. By establishing a robust existence theorem for right Bousfield localizations in the semimodel context, the work opens the door to systematic constructions of homotopy limits in settings previously inaccessible to model‑categorical techniques. Potential applications include the study of parametrized spectra, derived algebraic geometry over non‑Noetherian bases, and the development of “dual” localization theories where one inverts colimit‑type data rather than limit‑type data. The results also suggest a pathway toward a unified theory of left and right localizations within the flexible semimodel paradigm, which may simplify many constructions in modern homotopy theory.