An equivalence relation and groupoid on simplicial morphisms

In this work, we examine the simplicial morphisms homotopies. Firstly, we give the homotopy of free simplicial morphisms and later show that this homotopy relation indicates an equivalence. Then we obtain a groupoid, the simplicial morphisms between two fixed 1-truncated simplicial algebras (with free domain) are the objects, and the homotopies between this simplicial morphisms are the morphisms of this groupoid.