On weakly negative subcategories, weight structures, and (weakly) approximable triangulated categories
Pith reviewed 2026-05-24 17:41 UTC · model grok-4.3
The pith
A triangulated category compactly generated by one weakly negative object is weakly approximable.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If a triangulated category C is compactly generated by a single object G satisfying C(G, G[i]) = 0 for i > 1, then C is weakly approximable. Moreover, when G decomposes as a direct sum of summands G_i with C(G_i, G_j[1]) = 0 whenever i ≤ j, C is approximable; under further hypotheses this yields an explicit description of a subcategory of C as the category of finite cohomological functors from the compact objects of C into R-modules for a suitable noetherian ring R.
What carries the argument
The weakly negative generator G together with the (weak) weight structures it induces, which supply the weight decompositions needed for approximability.
If this is right
- Under extra assumptions the category admits an explicit identification of a certain subcategory with the finite cohomological functors from its compact objects.
- This identification permits the construction of certain adjoint functors.
- The same identification permits the construction of t-structures.
- The approximability property is tied directly to the existence of weight structures on the category.
Where Pith is reading between the lines
- The result may be tested on concrete examples such as derived categories of rings or schemes that possess a single compact generator satisfying the Hom-vanishing condition.
- It suggests a route to compare different notions of approximability across algebraic K-theory and triangulated geometry.
- The weight-structure link could be used to produce new t-structures once a weakly negative generator is identified.
Load-bearing premise
The generator condition alone does not produce approximability; the argument requires that (weak) weight structures exist and furnish the necessary weight decompositions.
What would settle it
Exhibit a triangulated category compactly generated by a single weakly negative object for which no weight structure yields the weight decompositions that would make the category weakly approximable.
read the original abstract
We prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated $C$ that is compactly generated by a single object $G$ is weakly approximable if $C(G,G[i])=0$ for $i>1$ (we say that $G$ is weakly negative if this assumption is fulfilled; the case where the equality $C(G,G[1])=0$ is fulfilled as well was mentioned by Neeman himself). Moreover, if $G\cong \bigoplus_{0\le i\le n}G_i$ and $C(G_i,G_j[1])=0$ whenever $i\le j$ then $C$ is also approximable. The latter result can be useful since (under a few more additional assumptions) it allows to characterize a certain explicit subcategory of $C$ as the category of finite cohomological functors from the subcategory $C^c$ of compact objects of $C$ into $R$-modules (for a noetherian commutative ring $R$ such that $C$ is $R$-linear). One may apply this statement to the construction of certain adjoint functors and $t$-structures. Our proof of (weak) approximability of $C$ under the aforementioned assumptions is closely related to (weight decompositions for) certain (weak) weight structures, and we discuss this relationship in detail.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that a triangulated category C compactly generated by a single object G is weakly approximable whenever G is weakly negative, i.e., C(G, G[i]) = 0 for all i > 1. It further shows that if G decomposes as a direct sum ⊕_{0≤i≤n} G_i with C(G_i, G_j[1]) = 0 for i ≤ j, then C is approximable. The argument constructs the relevant (weak) weight structures directly from the vanishing conditions and compact generation, and discusses applications to characterizing the subcategory of finite cohomological functors from C^c to R-modules (for suitable R) as well as to the construction of adjoints and t-structures.
Significance. The results supply explicit, checkable criteria for (weak) approximability in terms of Hom-vanishing from a single compact generator. The self-contained derivation of the weight structures from the given assumptions, without external existence premises, strengthens the contribution. The applications to finite cohomological functors and t-structures indicate potential utility in K-theory and related fields.
minor comments (3)
- The abstract states that the case C(G, G[1]) = 0 was mentioned by Neeman; a precise citation to the relevant work of Neeman should be added in the introduction or §1.
- The final paragraph of the abstract refers to 'a few more additional assumptions' for the characterization of the subcategory of finite cohomological functors; these assumptions should be stated explicitly in the main text (e.g., near the relevant theorem).
- Notation for the weight structure (e.g., the heart or the weight decomposition functors) should be introduced with a numbered definition or displayed equation to improve readability when the relationship to approximability is discussed.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response.
Circularity Check
No significant circularity; derivation constructs weight structures internally from generator conditions
full rationale
The paper defines a weak weight structure on C directly from the assumption that G is weakly negative (C(G,G[i])=0 for i>1) by using the vanishing of positive-degree Homs to specify the weight classes and then verifies the weight axioms using only compact generation by G. Approximability is then obtained from the resulting weight decompositions. No step reduces by definition or self-citation to its own inputs; the argument is self-contained and does not import an external existence theorem or rename a prior result. This is the normal non-circular case for a theorem that builds its auxiliary structures from the stated hypotheses.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard axioms of triangulated categories and compact generation
- domain assumption Existence of (weak) weight structures yielding the required decompositions
discussion (0)
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