A heap is a set \(H\) together with an associative ternary operation \([-,-,-]:H\times H\times H \to H\), satisfying the Mal’cev identities: \[ [[a, b, c], d, e] = [a, b, [c, d, e]]\ \ \ \text{and}\ \ \ [a, b, b] = a = [b, b, a], \] for all \(a,b,c,d,e\in H\). A heap \(H\) is called abelian if \([a, b, c] = [c, b, a]\) for all \(a,b,c\in H\). A truss is an abelian heap \(T\) together with an associative binary operation that distributes over the heap operation, that is: \[ s[t,t',t'']=[st,st',st'']\ \ \ \text{and}\ \ \ [t,t',t'']s=[ts,t's,t''s], \] for all \(s,t,t',t''\in T\). A left \(T\)-module is an abelian heap \(M\) together with an associative left action \(T\times M\to M\) (by \((t,m)\mapsto t\cdot m\)) of \(T\) on \(M\) that distributes over the heap operation.
In this paper, categorical aspects of the theory of modules over trusses are studied. It is defined the tensor product of modules over trusses by its universal property and it is presented an explicit construction that confirms the existence of the tensor product for all modules. It is shown, as in the case of modules over rings, tensoring with a bimodule over trusses defines a functor between the corresponding categories of modules. Moreover, truss versions of the Eilenberg-Watts theorem and Morita equivalence are given. Projective and small-projective modules over trusses are defined and their properties studied.