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Dilatations of categories, via their lean formalization

This paper presents a complete formalization in Lean 4 of the theory of category dilatations—a construction that modifies a category by forcing specific morphisms to factor uniquely through given maps—along with a systematic dictionary linking the mathematical theorems to their corresponding Lean declarations.

Original authors: Arnaud Mayeux

Published 2026-08-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Arnaud Mayeux

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 the vast landscape of mathematics not as a collection of isolated islands, but as a giant, interconnected city. In this city, Category Theory is the master mapmaker. It doesn't care about the specific details of the buildings (like whether they are made of brick or wood); instead, it cares about the roads connecting them and the rules for traveling between them. These "buildings" are called objects, and the "roads" are morphisms (or arrows).

Sometimes, mathematicians want to change the rules of the city to make travel easier. A classic trick is localization. Imagine a road that is currently a dead end or a toll booth that blocks traffic. Localization is like magically turning that road into a two-way street or removing the toll booth entirely, allowing you to travel backward or pass through freely. It's a powerful tool used everywhere, from algebra to geometry.

But what if you don't want to remove the road entirely? What if you just want to make it so that certain specific deliveries can pass through, while keeping the rest of the traffic rules intact? This is where dilatation comes in. Think of it as a "refined" version of localization. Instead of opening the whole gate, you build a special, narrow bypass lane that only allows specific packages (morphisms) to pass through a specific door, and only if they are accompanied by a specific key (a "sieve"). It's a more precise, surgical operation than the blunt force of standard localization.

Why does anyone care? Because these mathematical structures are the underlying code for how we understand shapes, spaces, and even the logic of computer programs. If we can prove these rules work perfectly, we can build more reliable software and solve complex problems in physics and engineering. However, human math is prone to tiny, invisible errors—a missing "if" or a slightly vague assumption. That's why this paper is special: it doesn't just write down the math; it forces a computer to check every single step, line by line, to ensure the logic is unbreakable.

The Paper: A Digital Blueprint for Mathematical Surgery

This paper, titled "Dilatations of Categories, Via Their Lean Formalization," is a report on a massive project where mathematician Arnaud Mayeux took a published mathematical theory about these "refined road rules" (dilatations) and translated it entirely into a language a computer can understand and verify. The computer tool used is called Lean 4, and it lives inside a giant library of verified math called Mathlib.

Think of the original mathematical paper as a set of architectural blueprints drawn by hand. They look correct, and other architects have nodded along, but there might be a tiny smudge on the paper or a step that was "obvious" to the human eye but actually skipped a crucial detail. Mayeux's job was to take those blueprints and rebuild them in a digital 3D modeling software that cannot make a mistake. If the math doesn't fit together perfectly, the software refuses to compile the code.

The Main Discovery: A New Way to Build
The paper's biggest finding isn't just that the math is correct; it's how the math was built. In the original theory, a "dilatation" was described as a collection of "fractions" (like $n/d$ ) glued together in a specific way. Doing this by hand is messy, like trying to build a house by stacking individual bricks one by one and checking if the wall is straight every time.

Mayeux's formalization took a different, smarter route. Instead of stacking bricks, they built a "skeleton" first—a free category (a raw, unconnected framework)—and then used a computer-generated "quotient" to snap the pieces together according to the rules. This approach is like using a 3D printer that knows the laws of physics: you don't have to manually check if the wall is straight; the printer guarantees it because the rules are built into the machine. This method allowed the team to prove the "universal property" of dilatations (the rule that says this is the only way to build this specific bypass) with absolute certainty.

The Plot Twist: When the Original Paper Had a Glitch
Here is where the story gets interesting. Because the computer is so strict, it found two places where the original published paper was slightly off.

The "Regular" Trap: In one section, the original paper claimed that a certain mathematical operation (combining two dilatations) always worked perfectly, like a magic trick that never fails. The computer, however, said, "Wait a minute. This only works if you add a specific extra condition." The formalization showed that without this extra condition, the magic trick fails. The paper didn't say the original math was useless, but it proved that the original claim was too broad. It's like saying "All birds can fly" until you realize penguins exist; the paper had to add a "penguin exception" to the rule to make it true.
The Ring vs. Category Mix-up: The paper also compared these category rules to rules for "commutative rings" (a type of algebra). The original paper suggested that a certain rule worked for both. The computer found a specific, tiny counterexample—a little mathematical puzzle with just two objects and a few arrows—where the rule worked for rings but completely broke for categories. It's like discovering that a bridge design that works for cars (rings) would collapse if you tried to drive a bicycle (categories) over it. The paper explicitly rules out the idea that the two theories are identical in this regard.

The "Codilatation" Shortcut
The paper also introduces a clever trick called "codilatation." Instead of writing a whole new book of rules for the opposite direction (where arrows point backward), the formalization simply said, "Let's flip the map upside down." By using the computer's ability to instantly swap "left" and "right," the team proved the rules for the backward direction without writing a single new proof. It's like realizing that if you know how to drive forward, you already know how to drive backward if you just turn the steering wheel the other way.

The Bottom Line
This paper is a triumph of "formalized mathematics." It proves that the theory of dilatations is solid, but it also acts as a quality control inspector, finding and fixing the tiny cracks in the original theory that human eyes missed. It shows that when you translate complex math into a language a computer understands, you don't just get a verification; you get a clearer, more precise understanding of the math itself. The paper concludes that while the theory is robust, it requires more careful conditions than previously thought, and it provides a complete, machine-checked dictionary for anyone who wants to use these "refined road rules" in the future.

Technical Summary: Dilatations of Categories, via Their Lean Formalization

Problem and Context
The paper addresses the formalization of the theory of "dilatations" of categories, a construction that refines the classical localization of categories. While classical localization inverts a collection of morphisms $\Sigma$ universally, dilatations introduce a more nuanced operation: given a "center" consisting of pairs $(d_i, N_i)$ where $d_i$ is a morphism and $N_i$ is a sieve over its codomain, the dilatation forces every morphism in $N_i$ to factor uniquely through $d_i$ . This construction generalizes the localization of commutative rings (specifically Néron blowups) to the categorical setting. The primary challenge addressed is the rigorous machine-checking of this theory, including the verification of its universal property, structural operations (restricting, shrinking, combining centers), and its relationship to ring-theoretic dilatations.

Methodology
The work is a complete formalization of a published mathematical paper (referenced as [5]) using the Lean 4 proof assistant and the Mathlib library. The methodology adheres to a strict principle: the formalization is the primary source of truth. Where the informal exposition and the formal development diverge, the formal version dictates the mathematical content, and discrepancies are documented as corrections or clarifications.

Key methodological choices include:

Construction via Quotients: Unlike the original paper, which constructs dilatations directly as equivalence classes of "fraction sequences," the formalization builds the dilatation $C'$ in two stages. First, it constructs a free category (a path category) on a quiver of generators (original morphisms and "fraction" edges). Second, it quotients this free category by a congruence relation defined by agreement in the raw localization $C[\Sigma^{-1}]$ . This approach leverages existing Mathlib infrastructure for path categories and quotient categories to avoid re-proving associativity and well-definedness from scratch.
Universal Property as the Core: The theory is organized around the universal property of the canonical functor $\Theta: C \to C'$ . Existence and uniqueness of the induced functor $F': C' \to D$ are proved using induction on the generated category (Lemma 5.9), relying on the faithfulness of the comparison functor to the raw localization.
Dualization: The theory of "codilatations" is not re-proven but derived definitionally by transporting the theory of dilatations across the opposite category duality ( $C \leftrightarrow C^{op}$ ).

Key Contributions and Results
The formalization produces several specific results and structural insights:

Universal Property (Theorem 5.8): The paper establishes the universal property of dilatations: a functor $F: C \to D$ factors uniquely through $\Theta$ if and only if $F$ is $\Sigma$ -regular (the localization of $D$ at the images of $\Sigma$ is faithful) and the sieves $N_i$ are preserved under $F$ in a specific sense.
Structural Operations: The formalization rigorously verifies operations on centers:
- Restriction: Restricting a center to a sub-family of indices yields a canonical functor between dilatations.
- Shrinking: Shrinking sieves within a center yields a faithful functor.
- Combination: Combining two centers is analyzed. The formalization reveals that the associativity of dilatations (dilating by $Z$ then by $W$ is isomorphic to dilating by $Z \cup W$ ) holds unconditionally for one direction but requires an explicit regularity hypothesis for the other direction.
Comparison with Rings: The paper formalizes the comparison between categorical dilatations and the dilatation of commutative rings. It demonstrates that the direct translation of ring dilatations (indexed directly by the original set of ideals) fails for categories due to the lack of additive closure in composition. The formalization corrects this by reindexing the center via exponent profiles, establishing an isomorphism between the categorical dilatation and the ring dilatation $A[M]$ .
Counterexamples and Corrections:
- Proposition 3.15 Correction: The formalization identifies a gap in the proof of [5, Proposition 3.15] regarding the regularity of the composite functor in the two-stage dilatation. The result is shown to hold only conditionally on an explicit regularity hypothesis, which is not automatically satisfied in general categories (though it is for rings).
- Proposition 5.1 Correction: The paper provides a counterexample showing that the identification of the categorical dilatation with the ring dilatation fails if the center is not reindexed correctly.
- Fact 5.2 Failure: The formalization constructs a finite counterexample showing that a characterization valid for rings (that every sub-algebra of a localization arises as a dilatation) fails for categories. Specifically, a subcategory of a localization may be faithful but not isomorphic to any dilatation of the original category.

Significance
The paper claims significance primarily as a step toward "indexed mathematics": a machine-verifiable, continuously updated corpus of mathematical knowledge. By formalizing a complete paper, including the discovery of minor errors and necessary clarifications in the original text, the work serves as:

A Verification: It confirms the correctness of the theory of dilatations of categories, while pinpointing exactly where the original proofs were insufficient.
Training Material: It provides a dataset of informal statements paired with machine-checked formal counterparts, intended for training AI systems in mathematics.
Methodological Demonstration: It illustrates how formalization can restructure mathematical arguments (e.g., using quotient categories instead of explicit equivalence classes) to leverage existing library infrastructure, thereby reducing the burden of proof and exposing hidden assumptions.

The work does not propose new applications of dilatations but rather solidifies the theoretical foundation of the existing theory, ensuring its reliability for future use in both human and machine-assisted mathematics.

The Paper: A Digital Blueprint for Mathematical Surgery

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