In 2017 a truss was defined. Thus one can say that the theory of trusses is new and not yet well-established. In recent years trusses start to gain attention due to their connections to ring theory and braces. Braces are closely related to solutions of set-theoretic Yang-Baxter equations, which can lead to applications of trusses in physics. In this thesis, we study connections among groups, heaps, rings, modules, braces and trusses. In the beginning, one can find a description in details of free heaps and coproducts of Abelian heaps. Both constructions are applied to describe a functor from the category of heaps to the category of groups. We establish a connection between unital near-trusses and skew left braces. We show that for a specific choice of congruence on a unital near-truss the quotient is a brace. We also prove that if one localises a regular unital near-truss without an absorber, the result is a skew left brace. In this thesis, one can find many small results on categories of heaps, trusses and modules over a truss. Methods to extend trusses to unital trusses and rings are presented. Then first one allows us to show that a category of modules over a truss is isomorphic with the category of modules over its extension to the unital truss. The second method establishes a deep connection between rings and trusses, i.e. every truss is an equivalence class of some congruence on some specific ring. We present the ring construction. Using this result, we introduce the definition of a minimal extension of a truss into a ring. We construct tensor product and free modules over trusses. The Eilenberg-Watts theorem for modules over trusses is stated and proven. Thus the Morita theory for modules over trusses is developed. The thesis is concluded with results on projectivity and decompositions through a product of the modules.
Morita theory and singularity categories
