Smooth categories in a 6 functor formalism and compact generation for nuclear categories in analytic geometry
Pith reviewed 2026-05-21 02:02 UTC · model grok-4.3
The pith
A rigid analytic variety is smooth if and only if its category of nuclear sheaves is smooth.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In the six-functor formalism supplied by condensed mathematics and analytic stacks, a rigid analytic variety is smooth if and only if the ∞-category of its nuclear sheaves is smooth. Compact generation of this category is related to algebraization of the variety, and these relations produce an example of a non-atomically generated but internally smooth category.
What carries the argument
The six-functor formalism for analytic stacks, which defines smoothness for ∞-categories of nuclear sheaves on rigid analytic varieties and carries the equivalence proof.
If this is right
- Smoothness of rigid analytic varieties can be checked by examining the smoothness of their nuclear sheaf categories.
- Compact generation of the nuclear sheaf category detects algebraization of the underlying rigid analytic variety.
- There exist internally smooth categories that are not atomically generated.
Where Pith is reading between the lines
- The same equivalence might be tested in other six-functor settings built from condensed objects.
- Categorical smoothness could serve as a new invariant for detecting algebraizability in analytic stacks.
- The example of a non-atomically generated smooth category invites classification of which smoothness properties require atomic generation.
Load-bearing premise
Condensed mathematics and analytic stacks supply a six-functor formalism in which smoothness is defined for categories of nuclear sheaves on rigid analytic varieties.
What would settle it
A single rigid analytic variety that is geometrically smooth yet whose nuclear sheaf category fails to be smooth, or the converse.
read the original abstract
In this paper, we study the notion of smooth $\infty$-categories within the framework of a six-functor formalism. By leveraging the theory of condensed mathematics and analytic stacks, we apply these results to demonstrate that a rigid analytic variety is smooth if and only if its associated category of nuclear sheaves is smooth. Furthermore, we relate the compact generation of the category of nuclear sheaves to the algebraization of the rigid analytic variety; these results are then employed to obtain an example of a non atomically generated but internally smooth category.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops the notion of smooth ∞-categories inside a six-functor formalism. Using condensed mathematics and analytic stacks, it proves that a rigid analytic variety X is smooth if and only if the ∞-category of nuclear sheaves on the associated analytic stack is smooth. It further relates compact generation of the nuclear sheaf category to algebraization of the rigid analytic variety and constructs an example of an internally smooth category that is not atomically generated.
Significance. If the central equivalence is established, the work supplies a categorical characterization of smoothness for rigid analytic varieties that could be useful for transferring geometric questions into the language of ∞-categories and six-functor formalisms. The explicit example of a non-atomically generated yet internally smooth category is a concrete contribution to the study of nuclear categories. The integration of condensed mathematics with analytic geometry is a positive feature when the technical interactions are fully verified.
major comments (1)
- The 'if' direction of the main equivalence (that categorical smoothness of the nuclear sheaf category implies geometric smoothness of the rigid analytic variety) is load-bearing. The argument must show that the six-functor formalism (f_!, f^*, etc.) together with nuclearity permits faithful reconstruction of the diagonal or the cotangent complex; without an explicit verification that nuclearity does not add or lose geometric information, the equivalence may require extra hypotheses on the base or on properness.
minor comments (2)
- The introduction would benefit from a short paragraph recalling the precise definition of nuclear sheaves and the analytic stack associated to a rigid analytic variety, for readers outside the condensed-mathematics community.
- Notation for the six functors and for the smoothness condition on ∞-categories should be made uniform across sections to avoid minor ambiguities.
Simulated Author's Rebuttal
We thank the referee for their thorough review and for identifying a key point in the proof of the main equivalence. We address the concern about the 'if' direction below and clarify how the six-functor formalism interacts with nuclearity. We believe the existing arguments suffice but will expand the exposition for clarity.
read point-by-point responses
-
Referee: The 'if' direction of the main equivalence (that categorical smoothness of the nuclear sheaf category implies geometric smoothness of the rigid analytic variety) is load-bearing. The argument must show that the six-functor formalism (f_!, f^*, etc.) together with nuclearity permits faithful reconstruction of the diagonal or the cotangent complex; without an explicit verification that nuclearity does not add or lose geometric information, the equivalence may require extra hypotheses on the base or on properness.
Authors: We appreciate this observation on the load-bearing nature of the 'if' direction. In Theorem 4.12, the proof proceeds by showing that smoothness of the nuclear sheaf category implies the existence of a cotangent complex via the six-functor formalism on the analytic stack. Specifically, the adjunctions f_! and f^* allow reconstruction of the diagonal morphism from the internal Hom and tensor structures preserved under nuclearity; nuclear sheaves are defined precisely so that they coincide with the dualizable objects in the condensed setting, ensuring no extraneous geometric data is introduced or lost. The reconstruction does not rely on properness or additional base hypotheses beyond those already stated for rigid analytic varieties. To make this reconstruction fully explicit, we will add a dedicated paragraph in Section 4.3 walking through the steps from categorical smoothness to the vanishing of the cotangent complex obstruction. We do not believe extra hypotheses are required, as the arguments are internal to the six-functor formalism for analytic stacks. revision: partial
Circularity Check
No significant circularity; central equivalence derived from external six-functor formalism
full rationale
The paper applies the six-functor formalism from condensed mathematics and analytic stacks to show that a rigid analytic variety is smooth precisely when its nuclear sheaves category is smooth in the categorical sense. This equivalence is presented as a result of the formalism's properties rather than a definitional identity or a fit to the target statement. The abstract and available description contain no equations or steps that reduce the smoothness criterion to a self-citation chain, a fitted parameter renamed as prediction, or an ansatz smuggled from prior author work. The additional results on compact generation and algebraization likewise appear to follow from the same external framework without collapsing the derivation to its inputs. The reader's assessment of score 1.0 is consistent with the absence of load-bearing self-referential reductions.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption A six-functor formalism can be defined and applied to categories of nuclear sheaves on rigid analytic varieties.
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 1.4 (Theorem 5.1). Let f:X→Sp(k) be a (classical) rigid analytic variety over k. Then X is smooth if and only if the ∞-category Nuc(X) is internally smooth in Pr^L_{Nuc(k)}.
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IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanalpha_pin_under_high_calibration unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Definition 2.21. Let C be a presentable and dualizable ∞-category over V. We say that C is internally smooth over V if the coevaluation map admits a V-linear right adjoint that preserves colimits.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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discussion (0)
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