Heaps of modules and affine spaces

Abstract

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules and modules over a ring and affine spaces are revealed and analysed. Contents 1. Introduction 2. Preliminaries and first results 2.1. Heaps and their morphisms 2.2. Trusses and modules 2.3. Isotropic and contractible modules 2.4. Abelian groups with a T -module structure 3. Heaps of modules 3.1. Heaps of modules. First properties 3.2. Isotropy and contractibility for heaps of modules 3.3. Crossed products by heaps of modules 4. Heaps of modules and affine spaces 4.1. Morphisms of heaps of modules as translations 4.2. Heaps of modules as affine spaces 5. Examples and applications 5.1. The Baer-Kaplansky Theorem for T -groups 5.2. Shelves, spindles, racks, quandles and the Yang-Baxter equation 5.3. Connections, splittings and contractions Acknowledgements Appendix A. Index of frequently used notation A.1. Categories A.2. Functors A.3. Special objects A.4. Other notations References

Lie trusses and heaps of Lie affebras

“…As is well-known (see e.g. [1]) but explored most recently in [5] an affine space can be defined as a set with two ternary operations. First, recall that an abelian heap [21], [3] is a set with a ternary operation…”

“…An affine space over a field F can be identified with a non-empty unital and absorbing heap of modules over F [5], i.e. with an abelian heap together with an operation…”

Heaps of Fish: arrays, generalized associativity and heapoids

“…The resulting graph is simple from the fact that morphism sets in a category are unique, undirected as a consequence of Proposition 2.2 and component-complete since all objects linked via some morphisms are also linked with all adjacent objects by composition and invertibility. A concrete instance of the heap approach to groupoids can be seen in the recent application of heap theory to the study affine structures [Bre+22;Brz22]. In its simplest form, the connection between heaps and affine structures works as follows: the point set of an affine space A can be regarded as the objects of a groupoid whose morphisms are the translations represented by the free vectors of the associated vector space V .…”

Lie trusses and heaps of Lie affebras

“…As is well-known (see e.g. [1]) but explored most recently in [5] an affine space can be defined as a set with two ternary operations. First, recall that an abelian heap [21], [3]…”

“…An affine space over a field F can be identified with a non-empty unital and absorbing heap of modules over F [5], i.e. with an abelian heap A together with an operation…”