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Kernel theorems for rigidly-compactly generated $\infty$ -categories

This paper establishes two representability theorems for functors between rigidly-compactly generated $\infty$ -categories by leveraging the interplay between compactness, dualizability, and coherence, thereby reformulating Grothendieck duality in terms of internal left adjoints and applying these results to $\mathbb{E}_\infty$ -ring spectra and algebraic geometry.

Original authors: Giovanni Rossanigo

Published 2026-06-10

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CC BY 4.0

Original authors: Giovanni Rossanigo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect working with a massive, infinite library of mathematical structures called $\infty$ -categories. These aren't just shelves of books; they are living, breathing universes where objects can be added, subtracted, and combined in complex ways.

In this paper, the author, Giovanni Rossanigo, is trying to solve a specific puzzle: How do we describe the "rules" (functors) that move things from one part of this library to another?

Specifically, he is looking at a special type of library called a "rigidly-compactly generated" category. Think of this as a library built from a finite set of "Lego bricks" (called compact objects). Even though the library is infinite, it's built entirely by snapping these finite bricks together in various ways.

Here is the breakdown of his discovery, using everyday analogies:

1. The Two Main Problems

The author wants to understand two types of "movers" (functors) between these libraries:

The "Backward" Mover: You have a rule that takes a "perfect" Lego brick and tells you what it looks like when transformed. Can we find a single "kernel" (a master blueprint) inside the destination library that explains this rule?
The "Forward" Mover: You have a rule that takes a "coherent" (well-behaved) structure and transforms it. Can we find a "perfect" Lego brick in the destination library that acts as the blueprint for this?

2. The Key Ingredients

To solve this, the author introduces three concepts that act like filters or lenses:

Compact Objects (The Bricks): These are the finite, manageable building blocks. In the real world, think of them as individual Lego bricks.
Pseudo-Coherent Objects (The Semi-Finished Models): These are structures that are "almost" finite. They might be huge, but if you look at them closely, they are built from a finite number of bricks in a specific, orderly way.
Coherent Objects (The Finished Models): These are the structures that are both "almost finite" and "bounded" (they don't stretch out forever in either direction). They are the perfectly assembled models.

3. The First Big Result: The "Backward" Blueprint

The Theorem: If you have a "quasi-proper" mover (a rule that respects the structure of the library and doesn't break the "semi-finished" models), then every rule that takes a "perfect brick" and turns it into a "semi-finished model" can be described by a single "semi-finished model" sitting in the destination library.

The Analogy:
Imagine you have a machine that takes a specific type of Lego brick (from Library A) and turns it into a complex, semi-finished car model (in Library B).
Rossanigo proves that there isn't a different machine for every single brick. Instead, there is one master blueprint (a specific semi-finished car model) sitting in Library B. If you know this blueprint, you can predict exactly what the machine will produce for any brick you feed it. You don't need to build a new machine for every brick; the blueprint does all the work.

4. The Second Big Result: The "Forward" Blueprint

The Theorem: This is trickier. If you want to describe a rule that takes "finished models" (coherent objects) and turns them into other "finished models," the blueprint isn't a finished model anymore. It's a perfect Lego brick.

However, there's a catch. The library of "finished models" is messy and doesn't always behave like a simple Lego set. To fix this, the author introduces a special condition called "Universal Descent."
Think of this as a "magic mirror" or a "translation device." If you can translate your messy library into a "regular" library (where finished models are just perfect bricks), you can find your blueprint.

The Analogy:
Imagine you have a rule that turns finished sculptures into other finished sculptures. Usually, you can't find a single "sculpture" that explains the rule. But, if you have a "magic mirror" (Universal Descent) that turns your complex sculptures back into simple Lego bricks, you can find a single Lego brick that acts as the blueprint. Once you have that brick, you can reconstruct the rule for the complex sculptures.

5. Why This Matters (The "Kernel" Idea)

The title mentions "Kernel Theorems." In math, a "kernel" is often the core piece of information that defines a whole system.

Old Way: To understand a complex transformation, you might need to list every single input and output.
Rossanigo's Way: You only need one single object (the kernel) to describe the entire transformation.

He proves that for these specific types of mathematical libraries, you can always shrink a massive, infinite rule down to a single, manageable "kernel" object.

6. Where This Applies

The author shows that this isn't just abstract theory; it works in real mathematical worlds:

Spectra: The study of shapes in high-dimensional space (like the building blocks of topology).
Schemes: The mathematical language used to describe geometric shapes in algebraic geometry (like curves and surfaces).
Spectral Algebraic Spaces: A modern, "quantum" version of those geometric shapes.

In all these cases, if you have a "proper" map (a well-behaved transformation between these spaces), you can use these theorems to find the "kernel" that defines how information flows between them.

Summary

Rossanigo has built a bridge between the "finite" (perfect bricks) and the "infinite" (complex structures). He proved that:

If you move from bricks to semi-finished models, the rule is defined by a semi-finished model.
If you move from finished models to finished models (under special conditions), the rule is defined by a brick.

This allows mathematicians to replace complicated, infinite descriptions with simple, finite blueprints, making it much easier to understand how these complex mathematical universes interact.

Technical Summary: Kernel Theorems for Rigidly-Compactly Generated $\infty$ -Categories

Problem Statement
The paper addresses the representability of linear functors between rigidly-compactly generated $\infty$ -categories, specifically focusing on "finitary" settings where the source and target are restricted to subcategories of "finite" objects defined via $t$ -structures. While classical kernel theorems (such as the equivalence between colimit-preserving functors and kernels in the ind-completion) are well-established for rigidly-compactly generated categories, they do not directly apply when the domain is restricted to pseudo-coherent or coherent objects. The central problem is to determine whether exact, linear functors defined on these finitary subcategories (e.g., $PCoh(B)$ or $Coh(B)$) can be represented by objects within the target category (or its dual) in a manner analogous to the classical Yoneda embedding. Specifically, the author seeks to establish equivalences between categories of functors and categories of "kernels" (objects in the target category) under specific finiteness conditions.

Methodology
The paper operates within the framework of presentable stable $\infty$ -categories ( $PrL_{st}$ ) and utilizes the language of commutative algebra objects in the symmetric monoidal category of small idempotent-complete stable categories ( $Cat^{perf}$ ). The methodology relies on three interacting notions of finiteness:

Compactness: Objects that preserve filtered colimits.
Dualizability: Objects admitting a dual, which is crucial for the rigidity of the monoidal structure.
Coherence/Pseudo-coherence: Objects defined via $t$ -structures, generalizing boundedness and finite generation in algebraic geometry.

The core technical approach involves:

Linear Yoneda Embeddings: Utilizing the $A$ -linear Yoneda embedding to relate modules over a rigid algebra $A$ to functors. The paper refines the standard equivalence $Ind(M^{op} \otimes_A N) \simeq Fun^{ex}_A(M, Ind(N))$ to the finitary setting.
Tor-Finite Categories: The author axiomatizes "tor-finite categories," which are rigidly-compactly generated categories equipped with an accessible, compatible, and complete $t$ -structure generated by compact objects of finite tor-dimension. This ensures that tensoring with compact objects preserves the finiteness properties of the $t$ -structure.
Universal Descent: To handle the representability of functors out of coherent objects (where the source is not generated by compact objects alone), the paper introduces "morphisms of universal descent." These are functors $h: B \to R$ to a "regular" category (where compact objects coincide with coherent objects) that allow the descent of properties from the target to the source via a finite-stage retract of the identity.
Projective Classes and Approximation: The proofs employ Neeman's theory of pseudo-compact objects and approximation systems. The author constructs projective classes generated by compact objects to approximate functors and utilizes the Barr-Beck theorem and comonadic descent to establish equivalences.

Key Contributions and Results
The paper establishes two main representability theorems (Kernel Theorems) for functors between tor-finite categories:

Functors out of $(B^\omega)^{op}$ (Theorem 1.0.1):
For a quasi-proper functor $f^L: A \to B$ (where the right adjoint preserves pseudo-coherent objects) and under the assumption that $B$ admits a finite generating set of compact objects that detects coconnective objects, there is an equivalence:
$PCoh(B) \simeq Fun^{ex}_{A^\omega}((B^\omega)^{op}, PCoh(A))$
$Coh(B) \simeq Fun^{ex}_{A^\omega}((B^\omega)^{op}, Coh(A))$
This result represents contravariant linear functionals on perfect objects with values in (pseudo-)coherent objects in terms of (pseudo-)coherent kernels.
Functors out of $Coh(B)$ (Theorem 1.0.2):
For a quasi-proper functor $f^L: A \to B$ , if $B$ admits a morphism of universal descent to a regular category $R$ , there is an equivalence:
$(B^\omega)^{op} \simeq Fun^{ex}_{A^\omega}(Coh(B), Coh(A))$
This result represents covariant linear functionals on coherent objects with values in coherent objects in terms of compact kernels (via the dual Yoneda embedding). This requires the descent condition because $Coh(B)$ is generally not generated by compact objects alone.

Applications and Examples
The abstract results are applied to several concrete geometric contexts:

$E_\infty$ -Ring Spectra: For a map $f: A \to B$ of connective $E_\infty$ -rings where $B$ is an almost compact $A$ -module, the theorems yield equivalences between categories of pseudo-coherent/coherent modules and functors between perfect modules.
Schemes: For a proper map $f: X \to Y$ of quasi-compact quasi-separated schemes with $Y$ noetherian, the paper recovers and generalizes results regarding $D^-_{coh}(X)$ and $D^b_{coh}(X)$ . Specifically, it establishes equivalences involving $Perf(Y)$-linear functors. If $X$ admits a quasi-perfect regular alteration, a duality for $Perf(X)^{op}$ is established.
Spectral Algebraic Spaces: The results extend to proper, locally almost of finite presentation morphisms of quasi-compact quasi-separated spectral algebraic spaces, relating pseudo-coherent and coherent objects to functors on perfect objects.

Significance and Claims
The author positions this work as a refinement of the "kernel theorem" for rigidly-compactly generated categories, specifically addressing the "finitary" case where the source is restricted to $t$ -structure-defined finite objects.

Generalization: The first theorem is presented as a generalization of Neeman's results (specifically [Nee25, Theorem 0.4.(i)]), while the second is described as a partial generalization of [Nee25, Theorem 0.4.(ii)].
Grothendieck Duality: The paper reformulates Grothendieck duality in terms of internal left adjoints, clarifying the relationship between quasi-proper functors and the preservation of pseudo-coherent objects.
Modesty: The author explicitly notes that the second theorem (Functors out of $Coh(B)$) is an "incomplete" generalization of Neeman's work, as it relies on the existence of a universal descent morphism to a regular category, a condition that is not always satisfied in the most general settings (e.g., for general spectral algebraic spaces). The paper does not claim to solve the representability problem for all coherent categories without such descent assumptions.
Motivation: The work is motivated by an informal question in [Nee25] regarding the parallelism between representability results in derived algebraic geometry (Ben-Zvi, Nadler, Preygel) and those in stable homotopy theory (Neeman). The paper aims to provide a unified abstract framework for these phenomena.