Every finite-dimensional compact connected abelian group has a canonical resolution as a quotient of a periodic locally compact group times a real vector space, and every morphism between such groups lifts to a product map.
Finite Rank Torsion Free Abelian Groups and Rings; Lecture Notes in Mathematics 931; Springer: Berlin/Heidelberg, Germany, 1982
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Structure of Finite-Dimensional Protori
Every finite-dimensional compact connected abelian group has a canonical resolution as a quotient of a periodic locally compact group times a real vector space, and every morphism between such groups lifts to a product map.