Unbounded Weight Structures: (Re)construction and Completion

Unbounded Weight Structures: Reconstruction and Completion

Overview

The paper "Unbounded Weight Structures: (Re)construction and Completion" (2605.00783) develops a robust theory for weight structures on stable $\infty$-categories, extending classical results for bounded weight structures to the unbounded, or non-compactly generated, setting. The authors establish a duality with the theory of (complete) $t$-structures, propose universal completion constructions for weight structures, and provide new characterizations of stable categories in terms of their weight hearts. A general construction principle is introduced, enabling the formation of weight structures in large presentable categories from small sets of generators, with applications spanning spectra, chain complexes, and module categories.

Weight Structures and Their Completions

A weight structure on a stable $\infty$-category $\mathcal{C}$ is determined by a subcategory $\mathcal{C}_{\ge 0}$, analogous but dual to a $t$-structure, designed for organizing "cellular" filtrations prevalent in algebraic and homotopical theories. In the bounded case, such structures are well studied: the minimal stable category generated by the heart $\mathcal{C}^{w\heartsuit}$ can be recovered from it, as demonstrated in works by Bondarko and Sosnilo.

The core contribution here is the extension beyond the bounded context, focusing on "complete" weight structures. The key notion is left completeness, mirroring left completeness in $t$-structures but with colimits replacing limits. The left completion $\lc{\mathcal{C}}$ of $\mathcal{C}$ is universal: any weight-exact functor from $t$0 to a left-complete weight category factors uniquely through $t$1. Complete weight structures are again determined by their weight heart, leading to a universal construction $t$2 from an additive category to a complete weight category, which precisely recovers classical homotopy categories of chain complexes.

Main Structural Theorems and Universal Constructions

The significant results include (numbers refer to those from the paper’s Introduction):

• Theorem A: For any stable category with a weight structure, the left completion $t$3 exists and is characterized by a universal property. Moreover, the weight heart undergoes idempotent completion through the process.
• Corollary A: The universal construction $t$4 from an additive category $t$5 produces a stable $t$6-category equivalent to the $t$7-categorical version of chain complexes, with the classical homotopy category $t$8 as its homotopy category. There is a right adjoint between weight hearts and complete weight categories.
• Theorem B: Any cocomplete stable category generated under colimits by a small set of objects admits a weight structure "compactly generated" from the set. This generalizes Bondarko's approach to broader cardinalities and enables constructing weight structures in new settings, including standard and novel ones on spectra, as well as on various module and derived categories.

Novel Examples and Recoverability from the Heart

Broad classes of algebraic and topological categories are shown to be "glued" or "assembled" from their weight hearts via explicit completion processes:

• The derived category $t$9 with the canonical weight structure has as heart the category of (arbitrary) projective abelian groups; the whole derived category is realized as $\infty$0.
• The category of bounded below spectra with finitely generated homotopy groups arises as the left completion of the subcategory of finite spectra.
• The entire stable category of spectra (as well as modules over a connective $\infty$1-ring $\infty$2) admits explicit complete weight structures, and the methods recover and generalize the standard weight structure on spectra and connect it with homological algebra over the Steenrod algebra.

A general theory of weak $\infty$3-structures is developed, relaxing the requirements of classical $\infty$4-structures and permitting "weak" completions. This is crucial for treating naturally occurring examples where full $\infty$5-structures are absent or non-adjacent, particularly for module categories and derived categories over non-discrete rings or abelian categories.

Implications and Applications

Theoretical

This work shows that (under suitable completeness assumptions) stable $\infty$6-categories with compatible weight and weak $\infty$7-structures can always be reconstructed from their heart via a two-stage process: left weight completion followed by right $\infty$8-completion, i.e., $\infty$9. This recasts many familiar stable categories—spectra, derived categories of projectives, and module categories—as completions from additive input, making the relationship between stable homotopy theory and algebra even more explicit.

The theory clarifies the limitations and possibilities for reconstructing or extending weight structures beyond compactly generated settings, and provides universal criteria for their existence and uniqueness.

Practical

The categorical constructions and completions introduced are highly constructive—objects in the completed category arise as colimits of explicit weight complexes or towers, making them amenable to concrete computation and potentially to algorithmic treatment in spectral algebraic geometry, motivic homotopy theory, and module-theoretic contexts.

Notable is the identification of the right weight completion of spectra with module categories over the Steenrod algebra, structuring the classical Adams spectral sequence in terms of weight completion; this has concrete consequences for calculations in stable homotopy theory.

Connections to Tate Objects and Exotic Weight Structures

An important application lies in the theory of Tate objects. The authors prove existence and characterize the weight heart for Tate categories under minimal assumptions and describe their completions—in particular, they show that even the Tate category of finite spectra does not generally admit an adjacent $\mathcal{C}$0-structure, only a weak one. This points to new categorical phenomena likely to influence the study of motives, condensed mathematics, and representation theory in derived settings.

The introduction of the "Anderson weight structure" on spectra, compactly generated by the Anderson dual of the sphere, demonstrates how the formalism uncovers previously inaccessible exotic weight structures, expanding the recognized landscape for potential applications.

Future Developments

Potential extensions include:

• The study of completion procedures for more general (not necessarily additive) settings, particularly in the context of motivic and equivariant homotopy theory.
• A systematic exploration of "exotic" weight structures facilitated by arbitrary sets of compact generators, particularly over base changes and in colored/multigraded scenarios.
• Applications to the theory of condensed or condensed Ind-objects, as indicated in relation to ongoing work on condensed mathematics.

Conclusion

This paper provides a comprehensive, categorical framework for understanding and generating unbounded (and complete) weight structures in stable $\mathcal{C}$1-categories. The authors’ work yields universal constructions, generalizes known results, and clarifies the deep relationship between additive hearts and the entire ambient stable category. These results open new pathways in the foundation and application of homotopical, algebraic, and representation-theoretic contexts, and lay groundwork for the systematic development of unbounded and exotic weight structures across modern mathematics.

References:

• "Unbounded Weight Structures: (Re)construction and Completion," (2605.00783)
• Sosnilo, V., "Some properties of weight structures," (Arnold-Roksandich et al., 2016)
• Bondarko, M., "Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)," Journal of K-Theory, 2010
• Lurie, J., "Higher Algebra," "Spectral Algebraic Geometry"