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A Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems

This paper constructs a categorical and 2-categorical realization of the categories of monoids and unital semigroups using Schützenberger categories and strict factorization systems, establishing 2-equivalences that provide a robust framework for studying Morita equivalence.

Original authors: Xavier Mary

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xavier Mary

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

In the vast landscape of modern mathematics, there is a field dedicated to understanding how things relate to one another, not just by their individual properties, but by the paths that connect them. This is the realm of category theory, a discipline that treats groups of objects and the rules for moving between them as the fundamental building blocks of structure. Within this field, a specific type of object called a monoid has long been a subject of intense study. A monoid is essentially a collection of items that can be combined in a specific order, where the order of combination matters but the grouping does not, and where there is a special "do nothing" item that leaves everything else unchanged. For decades, mathematicians have viewed these monoids as simple, single-pointed universes, where the items themselves are the only things that exist. However, this perspective often hides the rich internal geography of how these items interact. The question that drives recent inquiry is whether there is a different way to look at these structures that reveals their hidden connections more clearly, and whether this new view can solve old problems regarding when two seemingly different monoids are actually the same in a deeper, more functional sense.

A researcher has now constructed a new mathematical map that translates these single-pointed monoids into a different kind of landscape: a small world filled with many distinct points, where the items of the original monoid become the locations themselves. In this new world, the rules for moving between locations are governed by a precise system of paths that can be broken down into two distinct, non-overlapping types of steps. One type of step moves forward in a way that cannot be undone, while the other type moves in a way that cannot be repeated. The researcher proved that every monoid has a unique counterpart in this new world of points and paths, and conversely, that every world built with these specific rules of movement corresponds to exactly one monoid. This is not just a simple reshuffling of labels; it is a complete structural translation that preserves every detail of the original object. By building this bridge, the researcher has shown that the complex algebra of monoids is identical to the geometry of these specific path systems.

The power of this discovery lies in how it handles the concept of equivalence. In mathematics, two objects are often considered "the same" if one can be transformed into the other without losing any essential information. However, for monoids, there is a more subtle and powerful kind of sameness known as Morita equivalence. This concept, which has been difficult to pin down using traditional methods, describes a situation where two monoids might look completely different on the surface but function identically in the context of their larger mathematical environment. The new map created by the researcher acts as a perfect lens for this phenomenon. They demonstrated that when two monoids are Morita equivalent, their corresponding worlds of points and paths are connected by a special kind of reversible relationship. This relationship is not just a simple match; it involves a set of instructions that can move back and forth between the two worlds, transforming one into the other and back again without any loss of data.

To understand how this works, imagine the monoid as a single room where people can only move by following a single, rigid set of instructions. The new approach expands this room into a vast city where every person is a distinct building, and the instructions for moving between them are laid out on a grid. The researcher showed that the rules for navigating this city are so strict and well-defined that you can reconstruct the original single room perfectly from the city map. Furthermore, they found that the special "reversible" connections between two different cities correspond exactly to the deep functional equivalence between the original rooms. This means that if two monoids are Morita equivalent, their city maps are linked by a pair of guides who can lead you from one city to the other and back, proving that the two cities are, in a profound sense, the same place.

This work does more than just offer a new way to visualize old objects; it provides a rigorous proof that these two ways of looking at the world are fundamentally interchangeable. The researcher established that the process of turning a monoid into a city of paths and the process of turning a city of paths back into a monoid are perfect inverses of each other. They also extended this result to a slightly broader class of objects called unital semigroups, which are similar to monoids but have slightly different rules for their "do nothing" item. The findings confirm that the deep structural properties of these algebraic systems are best understood not by staring at the items themselves, but by observing the network of relationships that bind them together. By proving that the category of monoids is equivalent to this specific category of structured paths, the study offers a powerful new tool for mathematicians to classify and understand the hidden symmetries of algebraic systems, turning a difficult abstract problem into a clear, geometric reality.

Technical Summary: A Categorical Realization of the (2-)Category of Monoids via Schützenberger Categories and Strict Factorization Systems

Problem Statement
The paper addresses the categorical representation of monoids and unital semigroups. While it is standard to view a monoid $M$ as a one-object category (its "delooping" $BM$), the author seeks an alternative perspective. The goal is to construct a category equivalent to the category of monoids ( $\mathbf{Mon}$ ) and monoid homomorphisms, and subsequently the category of unital semigroups ( $\mathbf{Mons}$ ), using a "loop space" construction. This construction aims to associate a monoid with a small category whose objects are the elements of the monoid, equipped with specific structural properties that facilitate the study of Morita equivalence.

Methodology
The methodology relies on the interplay between semigroup theory and category theory, specifically utilizing Schützenberger categories and Strict Factorization Systems (SFS).

Strict Factorization Systems (SFS): The paper adopts the definition from Grandis, where an SFS on a category $\mathcal{C}$ is a pair of wide subcategories $(E, M)$ such that every morphism $f$ admits a unique decomposition $f = em$ with $e \in E$ and $m \in M$ . The author focuses on SFSs that are:
- Unital: Possessing a unit object $\zeta$ that is a strict initial point of $E$ and a strict terminal point of $M$ .
- Complete: Satisfying a commutation property allowing the completion of certain diagrams (related to the commutation of Green's preorders $\leq_L$ and $\leq_R$ ).
- Thin: Containing at most one morphism between any two objects.
Schützenberger Category Construction ( $D$ ): For a semigroup $S$ , the author utilizes the Schützenberger category $D(S)$ , originally defined by Costa and Steinberg.
- Objects: Elements of $S$ .
- Morphisms: Triples $(a, x, b)$ representing arrows $a \xrightarrow{x} b$ , where $x \in aS^1 \cap S^1b$ .
- Structure: $D(S)$ is shown to possess a canonical strict factorization system $(E, M)$ where $E$ -maps correspond to right multiplications and $M$ -maps to left multiplications. For a monoid $M$ , $D(M)$ is identified as the Freyd (or epi-monic) completion of the delooping category $BM$.
Reconstruction of Monoids ( $\Sigma$ ): Conversely, given a small category $\mathcal{A}$ with a unital, complete, and thin SFS (denoted $\mathbf{CatSFS}^*_{c,t}$ ), the author defines a binary operation on the set of objects $X = \mathcal{A}_0$ . For objects $a, b$ , the product $a * b$ is defined as the central object in the unique factorization of the unique map $a \xrightarrow{\zeta} b$ . This construction, denoted $\Sigma(\mathcal{A})$ , yields a monoid.
2-Categorical Extension: The paper extends these 1-categorical equivalences to 2-categories by introducing natural transformations.
- In the context of pointed functors, natural transformations are trivial.
- In the context of "semi-pointed" functors (which do not strictly preserve the unit but preserve idempotency), natural transformations correspond to conjugations (elements $\alpha$ in the target semigroup satisfying specific commutation relations with the homomorphisms).

Key Contributions and Results

Equivalence of Categories:
The paper establishes that the category of monoids $\mathbf{Mon}$ is equivalent to the category $\mathbf{CatSFS}^*_{c,t}$ (small categories with unital, complete, thin SFS, pointed at the unit).
- The functor $D: \mathbf{Mon} \to \mathbf{CatSFS}^*_{c,t}$ maps a monoid to its Schützenberger category.
- The functor $\Sigma: \mathbf{CatSFS}^*_{c,t} \to \mathbf{Mon}$ reconstructs the monoid from the category.
- Theorem 2.1 proves that $D \dashv \Sigma$ is an adjoint equivalence, with $\Sigma \circ D = \text{Id}_{\mathbf{Mon}}$ .
- A parallel result (Theorem 2.2) extends this equivalence to unital semigroups ( $\mathbf{Mons}$ ) and semi-pointed categories ( $\mathbf{CatSFS}_{u,c,t,s}$ ).
2-Equivalence and Conjugations:
By defining natural transformations between semi-pointed functors, the author constructs a 2-category $\mathbf{Mon}^{(2)}_s$ where 2-cells are conjugations.
- A conjugation $\alpha: f \Rightarrow g$ between homomorphisms $f, g: M \to M'$ is an element $\alpha \in M'$ such that $f(1)\alpha = \alpha = \alpha g(1)$ and $f(m)\alpha = \alpha g(m)$ for all $m$ .
- Theorem 3.3 establishes a strict 2-equivalence between $\mathbf{Mon}^{(2)}_s$ and the 2-category of Schützenberger categories with natural transformations.
Characterization of Morita Equivalence:
The primary result (Theorem 3.7) connects the 2-categorical structure to Morita equivalence.
- The paper proves that two monoids $M$ and $M'$ are Morita equivalent if and only if they are adjointly equivalent in the 2-category $\mathbf{Mon}^{(2)}_s$ .
- This is demonstrated by showing that an adjoint equivalence in $\mathbf{Mon}^{(2)}_s$ (involving invertible conjugations satisfying triangle identities) corresponds precisely to the existence of an idempotent $e$ such that $M = MeM$ and $M' \cong eMe$ , which is the standard characterization of Morita equivalence for monoids.

Significance
The paper claims that the 2-category $\mathbf{Mon}^{(2)}_s$ , constructed via Schützenberger categories and strict factorization systems, provides the "good" framework for studying Morita equivalence of monoids. Unlike the standard delooping approach (one-object categories), this realization treats monoids as categories where objects are elements, allowing the structural properties of Morita equivalence (specifically adjoint equivalences) to emerge naturally from the interplay of conjugations and factorization systems. The work unifies the categorical realization of monoids with the algebraic theory of Morita equivalence, offering a direct proof of the correspondence without relying on topos theory.

Technical Summary: A Categorical Realization of the (2-)Category of Monoids via Schützenberger Categories and Strict Factorization Systems

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