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Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories

This paper investigates the category of atomic monoids ( $\mathsf{AtoMon}$ ), establishing its local finite presentability and specific factorization properties while demonstrating it is not a regular category, and further constructing key adjunctions and lifting torsion theories from groups to this setting.

Original authors: Federico Campanini, Laura Cossu

Published 2026-07-28

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Federico Campanini, Laura Cossu

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 a world where everything is built from tiny, indivisible Lego bricks. In mathematics, there's a whole branch called "category theory" that studies how different shapes and structures fit together, not just by their size or color, but by how they connect and transform into one another. Think of it as the ultimate rulebook for how things relate. One specific shape in this world is a "monoid," which is just a fancy name for a collection of things you can combine (like multiplying numbers) that always has a "do nothing" piece (like the number 1).

Now, imagine a special club of these monoids called "atomic monoids." In this club, every single piece that isn't the "do nothing" piece can be broken down into a set of those tiny, indivisible Lego bricks, which mathematicians call "atoms." Just like you can take apart a complex Lego castle to see the individual bricks, mathematicians study how these atomic monoids break down. A key feature of this club is that these breakdowns are often not unique—a single piece might be assembled from different combinations of bricks in different ways. The big question this paper tackles is: "What are the rules of the game when we treat these atomic monoids as a family of shapes that can be stretched, squished, and glued together?" It turns out that while they follow some familiar rules, they also have some very surprising quirks that break the standard rulebook.

The Atomic Club: A New Kind of Math Playground

In this paper, Federico Campanini and Laura Cossu invite us into the Category of Atomic Monoids (or AtoMon for short). Think of AtoMon as a massive, bustling city where every building is an atomic monoid, and the roads connecting them are special maps that only let you pass through if you respect the "atoms" (the indivisible bricks). The authors want to know: Is this city well-organized? Does it follow the standard laws of mathematical architecture, or is it a chaotic mess with its own unique physics?

The City is Well-Organized (Locally Finitely Presentable)

First, the authors prove that AtoMon is a very tidy city. In math-speak, they show it is "locally finitely presentable." To use an analogy, imagine you want to build any building in this city. The authors prove that you don't need to invent new materials from scratch every time. Instead, you can build any complex structure by gluing together a specific, finite set of "starter kits" (called compact objects).

These starter kits are special because they are small enough to be described with a finite list of rules (generators and relations), yet they are powerful enough to construct any other atomic monoid in the city. It's like saying that no matter how complex a Lego castle you want to build, you can always make it by combining a specific, finite collection of basic Lego sets. The authors even found exactly which sets these are: the free monoids, the infinite cyclic groups, and some specific monoids made by forcing two long words to be equal. This means the city is predictable and manageable.

The Broken Mirror: Why the City Isn't "Regular"

Here is where things get tricky. In the world of category theory, there is a concept called a "regular category." You can think of a regular category as a place where if you take a perfect, smooth path (a "regular epimorphism") and look at it through a mirror (a "pullback"), the reflection is also a perfect, smooth path. It's a rule of consistency: if something works one way, it should work the same way in a slightly different context.

The authors discovered that AtoMon breaks this rule. They constructed a specific example of a "perfect path" (a regular epimorphism) that, when reflected through a mirror, turns into a path that is still a valid, surjective map, but it loses its "perfect" status (it is no longer a regular epimorphism).

The Analogy: Imagine you have a machine that perfectly sorts red and blue marbles into two separate bins. This machine works perfectly in the main room. But if you move this machine into a side room (the "pullback"), the machine still sorts the marbles and sends them all through (it's still a surjective map), but the sorting mechanism itself is now flawed or "bumpy" in a way that violates the specific rules of the main room. It's not that the machine stopped working; it's that it stopped being the kind of machine that the rules require.
The Result: Because this "flawed reflection" happens, AtoMon is not a regular category. This is a big deal because it means AtoMon cannot be described as a simple "variety of universal algebras" (a standard, well-behaved type of mathematical structure). It has a personality of its own that refuses to follow the standard script.

However, the authors didn't just say "it's broken." They showed that every map in AtoMon can still be split into a "perfect path" part and a "one-way street" part (a (regular epi, mono)-factorization). They just proved that the "perfect path" part isn't stable enough to survive a mirror test.

New Tools: The "Atomization" Machine

The paper also introduces some cool new tools (functors) that act like machines transforming one type of object into another.

The Group-of-Units Machine: Every atomic monoid has a special group of "invertible" pieces (units). The authors show you can extract this group, and they built two machines to go back and forth between the world of groups and the world of atomic monoids. One machine adds a "trivial" layer to a group to make it an atomic monoid, and another strips away the non-group parts.
The Atomization Machine: This is perhaps the most creative tool. The authors built a machine that takes any ordinary monoid (even one that isn't atomic) and forces it to become an atomic monoid. It does this by adding a "skeleton" of atoms and a "trash can" for everything else. It's like taking a pile of random junk and forcing it to organize itself into a structure where every piece is either a fundamental atom or a unit, with a special "zero" bucket for the rest. This machine is the "right adjoint" to the inclusion of atomic monoids, meaning it's the best possible way to turn a messy monoid into a tidy atomic one.

Lifting Torsion Theories: The "Good vs. Bad" Filter

Finally, the authors tackle a concept called "pretorsion theories." In simple terms, this is a way to divide a category into two camps: "Good" objects and "Bad" (or trivial) objects, with a rule for how they interact.

They took a known way of dividing Groups into "torsion" (bad) and "torsion-free" (good) groups.
They then figured out how to lift this division up into the world of Atomic Monoids.
The Result: They created a new split in AtoMon. One side is the group of units (the "good" groups), and the other side is the "reduced" monoids (those with no units other than the identity). They proved that every atomic monoid can be broken down into a "group part" and a "reduced part" in a very specific, structured way. This is like having a universal filter that can separate the "group-like" behavior from the "purely atomic" behavior in any structure you throw at it.

The Takeaway

This paper doesn't just say "atomic monoids exist." It maps out their entire neighborhood. It proves they are built from a finite set of Lego kits, shows that they have a weird quirk where their "perfect paths" break when mirrored (becoming valid but non-regular maps), and provides a toolkit to transform any monoid into an atomic one. Most importantly, it shows that while AtoMon is a rich and complex mathematical world, it is not a "regular" one—it has its own unique, slightly chaotic logic that makes it fascinatingly different from the standard algebraic structures we usually study. The authors have successfully drawn the map, showing us exactly where the rules hold and where they break.

Technical Summary: Categorical Algebra of Atomic Monoids

Problem Statement
The paper investigates the categorical properties of the category $\mathbf{AtoMon}$ , whose objects are atomic monoids and whose morphisms are atom-preserving monoid homomorphisms. While the category was previously introduced to study the arithmetic behavior of universal constructions in factorization theory (specifically products, coproducts, and their associated invariants), its broader structural properties within categorical algebra remained largely unexplored. The authors aim to determine whether $\mathbf{AtoMon}$ satisfies standard categorical properties such as local finite presentability, regularity, and the existence of specific adjunctions, and to explore its relationship with torsion theories.

Methodology
The authors employ a combination of explicit construction and abstract categorical reasoning:

Structural Analysis: They analyze the behavior of limits and colimits in $\mathbf{AtoMon}$ , noting that while colimits coincide with those in the category of monoids ( $\mathbf{Mon}$ ), limits generally do not.
Compact Object Identification: To establish local finite presentability, the authors construct a specific set of "compact" objects (finitely presented atomic monoids) and demonstrate that they form a strong generator.
Counterexample Construction: To test regularity, the authors explicitly construct a regular epimorphism and analyze its pullback. They utilize kernel pairs and congruences internal to $\mathbf{AtoMon}$ (specifically the congruence $\theta_f$ generated by pairs of atoms and units) to distinguish between regular and non-regular morphisms.
Adjunction Construction: The authors define specific functors (group of units, atomization) and construct their adjoints by defining explicit mappings on units, atoms, and non-unit non-atoms.
Lifting Theorems: They develop a general criterion for lifting pretorsion theories along a monocoreflection, applying it first to the specific case of $\mathbf{AtoMon}$ and then to a distinct example involving pointed quivers.

Key Contributions and Results

Local Finite Presentability: The paper proves that $\mathbf{AtoMon}$ is locally finitely presentable (Theorem 2.6). This is achieved by exhibiting a strong generator $\mathcal{G}$ consisting of compact objects. The set $\mathcal{G}$ includes the free monoid on one generator, the infinite cyclic group, and specific reduced atomic monoids defined by relations between words of length at least two. Furthermore, the authors characterize the compact objects of $\mathbf{AtoMon}$ precisely as the atomic monoids that are finitely presented as ordinary monoids (Theorem 2.7).
Failure of Regularity: The authors demonstrate that $\mathbf{AtoMon}$ is not a regular category.
- They establish that every morphism in $\mathbf{AtoMon}$ admits a canonical $(regular\ epi, mono)$ -factorization (Proposition 3.3).
- However, they construct a specific regular epimorphism $p: H \to Q$ whose pullback along a monomorphism $j: J \to Q$ yields a morphism $\rho: P \to J$ that is surjective but not a regular epimorphism (Proposition 3.7).
- Consequently, $\mathbf{AtoMon}$ is not equivalent to a variety of universal algebras. The paper notes that while not regular, it is a finitary essentially algebraic category (equivalent to models of a finitary finite-limit theory).
- The authors also describe the regular completion of $\mathbf{AtoMon}$ in the sense of Carboni and Vitale, showing that certain objects in the completion are not isomorphic to images of atomic monoids under the canonical embedding.
Adjunctions:
- Group of Units: The functor $(\cdot)^\times: \mathbf{AtoMon} \to \mathbf{Grp}$ is shown to have both a left adjoint (the inclusion functor $\iota$ ) and a right adjoint $I$ (Proposition 4.1). The right adjoint $I(G)$ constructs an atomic monoid from a group $G$ by adjoining a zero and an atom.
- Atomization: The inclusion functor $U: \mathbf{AtoMon} \to \mathbf{Mon}$ is shown to have a right adjoint $A: \mathbf{Mon} \to \mathbf{AtoMon}$ (Proposition 4.2). This "atomization" functor constructs an atomic monoid $A(M)$ from a monoid $M$ by embedding $M$ into a structure with a terminal object component.
Pretorsion Theories: Since $\mathbf{AtoMon}$ is not a pointed category, it cannot admit strict torsion theories. The authors lift torsion theories from $\mathbf{Grp}$ to pretorsion theories in $\mathbf{AtoMon}$ .
- Specifically, they show that if $(\mathcal{T}, \mathcal{F})$ is a torsion theory in $\mathbf{Grp}$ , the pair $(\tilde{\mathcal{T}}, \tilde{\mathcal{F}})$ forms a pretorsion theory in $\mathbf{AtoMon}$ , where $\tilde{\mathcal{T}} = \mathcal{T}$ and $\tilde{\mathcal{F}}$ consists of atomic monoids whose group of units lies in $\mathcal{F}$ .
- A notable instance is the pretorsion theory $(\mathbf{Grp}, \mathbf{RedAtoMon})$ , where $\mathbf{RedAtoMon}$ is the subcategory of reduced atomic monoids (those with trivial unit groups).
- This result is generalized in Theorem 5.6 to any monocoreflective subcategory, providing a criterion to lift pretorsion theories.

Significance and Claims
The paper claims to provide a comprehensive structural analysis of $\mathbf{AtoMon}$ , moving beyond its arithmetic applications to establish its place within categorical algebra.

It confirms that $\mathbf{AtoMon}$ is a well-behaved category in terms of presentability, allowing the application of general categorical tools.
It clarifies the limitations of the category by proving it is not regular, thereby distinguishing it from varieties of universal algebras and highlighting the specific arithmetic constraints imposed by the "atom-preserving" condition.
It establishes a bridge between the arithmetic of atomic monoids and the categorical structures of groups and general monoids through explicit adjunctions.
It offers a novel method for generating pretorsion theories in non-pointed categories by lifting them from pointed subcategories, illustrated by the relationship between groups and reduced atomic monoids.

The authors position these results as a necessary step to understand the "rich structure" of $\mathbf{AtoMon}$ as a category in its own right, providing the foundational categorical framework required to further study realization problems in factorization theory.