Enhancement for categories and homotopical algebra

We develop foundations for abstract homotopy theory based on Grothendieck’s idea of a “derivator”. The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory.

💡 Research Summary

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The paper develops a model‑independent foundation for abstract homotopy theory by building on Grothendieck’s notion of a derivator. Rather than relying on model categories or simplicial sets, the author constructs a framework that works entirely within ordinary category theory while still capturing the essential features of homotopical algebra.

The first part revisits basic categorical notions—objects, functors, adjunctions, limits, colimits, and additive structures—setting a uniform language for the later sections. A substantial portion is devoted to the Grothendieck construction and its generalizations: Cartesian functors, families of groupoids, and the interplay between fibrations and cofibrations in a higher‑dimensional setting.

Next, the author studies ordered structures in great detail. Partially ordered sets are examined through left‑closed embeddings, reflexive maps, and barycentric subdivisions. These concepts are then lifted to bi‑ordered sets and bico‑fibrations, providing a combinatorial backbone for later homotopical constructions.

The simplicial side of the theory is treated in depth. After recalling ordinals, nerves, and the Segal condition, the paper introduces Segal categories, special maps, and twisted 2‑simplicial expansions. Reedy categories, cellular Reedy categories, skeleta, and various subdivision techniques are developed to supply explicit resolutions needed for homotopical calculations.

Section 4 is the conceptual heart: abstract localization. The author defines relative categories (a category together with a distinguished class of morphisms) and weak equivalences, then constructs the universal localization functor satisfying the usual universal property. Particular attention is given to localization over the simplex category Δ, where simplicial expansions and regular Segal categories give concrete models. Model structures are introduced in a derivator‑compatible way, providing notions of cofibrant objects and explicit homotopies without invoking a full model category.

Section 5 introduces CW‑categories and semiexact families, extending the classical CW‑complex intuition to an abstract categorical setting. The Reedy construction, abundant C‑categories, relatively cofibrant functors, and CW‑augmentations are defined, and a Mayer‑Vietoris presentation theorem is proved for these objects. A representability criterion is established, showing that many familiar homotopical constructions (e.g., homotopy colimits) can be recovered inside this framework.

Section 6 bridges the gap between simplicial and ordered viewpoints by defining enhanced groupoids, Segal spaces, and complete Segal spaces. Augmented Segal families, cylinder axioms, weak excision, and barycentric dualization are proved, demonstrating that the enhanced structures behave as expected under homotopical operations.

Section 7 is the culmination: the theory of enhanced categories and functors. Reflexive families, truncation/unfolding procedures, and augmented families are introduced, leading to a definition of enhanced categories that simultaneously carry a categorical and homotopical layer. The author constructs enhanced cylinders, enhanced fibrations/cofibrations, an enhanced Grothendieck construction, and an enhanced Yoneda embedding. Limits, colimits, Kan extensions, and localizations are all re‑interpreted in this enhanced setting. Large‑category issues such as Karoubi envelopes, filtered colimits, accessible categories, and tame fibrations are also addressed, showing that the framework scales to the size of modern homotopy‑theoretic applications.

The introductory section provides a philosophical motivation, contrasting the “angel‑devil” metaphor for geometry versus algebra and arguing that homotopy theory is fundamentally a localization process. The author contends that existing approaches (model categories, simplicial sets) obscure the underlying categorical nature of homotopy, and that the derivator‑based enhancement presented here restores that clarity.

Overall, the paper offers a comprehensive, self‑contained categorical apparatus for homotopical algebra that avoids the technical baggage of model categories while preserving the ability to perform concrete constructions such as derived functors, homotopy limits, and localization. It promises to be a valuable foundation for future work on enhancements of derived categories of coherent sheaves, stable ∞‑categories, and other areas where a clean, model‑independent homotopy theory is desired.