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Heaps of modules: Categorical aspects

Simion Breaz, Tomasz Brzezinski, Bernard Rybolowicz, Paolo Saracco

TL;DR

The work unifies the algebra of heaps of modules over a truss $T$ with the module theory over the universal ring $R(T)$, proving equivalences between $T ext{-Mod}_ullet$ and $R(T) ext{-Mod}$, and between heaps of $T$-modules and affine $R(T)$-modules. By exploiting these equivalences, it describes limits, colimits, and exact sequences in the Barr-exact category of heaps of $T$-modules through well-known ring-theoretic mechanisms, including the Dorroh extension for unitalization. The paper provides explicit constructions for (co)limits (equalizers, products, quotients, coequalizers, coproducts, pushouts) and relates Barr-exactness to ring-theoretic exactness via a slice-category framework, thereby offering a coherent, algebraic framework for the categorical study of trusses and their modules. This synthesis clarifies how affine and module structures over rings capture the behavior of truss-based algebra, enabling a unified treatment of free objects, exactness, and categorical properties across both settings.

Abstract

Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and algebraic theories. In particular, it is shown that the category of groups with a compatible action of a truss T (also called pointed T-modules) is isomorphic to the category of modules over the ring R(T) universally associated to the truss. This is widely used in the explicit description of free objects. Next, it is proven that the category of heaps of modules over T is isomorphic to the category of affine modules over R(T) and, in order to make the picture complete, that (in the unital case) these are in turn equivalent to a specific subcategory of the slice category of pointed T-modules over R(T). These correspondences and properties are then used to describe explicitly various (co)limits and to compare short exact sequences in the Barr-exact category of heaps of T-modules with short exact sequences as defined previously.

Heaps of modules: Categorical aspects

TL;DR

The work unifies the algebra of heaps of modules over a truss with the module theory over the universal ring , proving equivalences between and , and between heaps of -modules and affine -modules. By exploiting these equivalences, it describes limits, colimits, and exact sequences in the Barr-exact category of heaps of -modules through well-known ring-theoretic mechanisms, including the Dorroh extension for unitalization. The paper provides explicit constructions for (co)limits (equalizers, products, quotients, coequalizers, coproducts, pushouts) and relates Barr-exactness to ring-theoretic exactness via a slice-category framework, thereby offering a coherent, algebraic framework for the categorical study of trusses and their modules. This synthesis clarifies how affine and module structures over rings capture the behavior of truss-based algebra, enabling a unified treatment of free objects, exactness, and categorical properties across both settings.

Abstract

Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and algebraic theories. In particular, it is shown that the category of groups with a compatible action of a truss T (also called pointed T-modules) is isomorphic to the category of modules over the ring R(T) universally associated to the truss. This is widely used in the explicit description of free objects. Next, it is proven that the category of heaps of modules over T is isomorphic to the category of affine modules over R(T) and, in order to make the picture complete, that (in the unital case) these are in turn equivalent to a specific subcategory of the slice category of pointed T-modules over R(T). These correspondences and properties are then used to describe explicitly various (co)limits and to compare short exact sequences in the Barr-exact category of heaps of T-modules with short exact sequences as defined previously.
Paper Structure (18 sections, 16 theorems, 101 equations)

This paper contains 18 sections, 16 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heaps and their morphisms
  4. Heaps and groups
  5. Trusses and their modules
  6. Trusses and rings
  7. Heaps of modules
  8. Heaps of modules over a truss and affine modules over a ring
  9. Pointed modules over a truss and modules over a ring
  10. Heaps of modules over a truss and affine modules over a ring
  11. Categorical aspects of heaps of modules
  12. Limits
  13. Colimits
  14. Modules
  15. Heaps of modules
  16. ...and 3 more sections

Key Result

Theorem 2.6

Let $M$ be a $T$-module and $N \subseteq M$ a non-empty sub-heap. The quotient $M/N$ is a $T$-module with the canonical map $\pi:M\to M/N$ being an epimorphism of $T$-modules if and only if $N$ is an induced submodule of $M$.

Theorems & Definitions (60)

  • Remark 2.1: Brz:par
  • Remark 2.2
  • Remark 2.3: BrBrRySa
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6: Brz:par
  • Definition 2.7: BrBrRySa
  • Remark 2.8
  • Example 2.9
  • Remark 2.10: ABR
  • ...and 50 more