For $V^\bullet$ a complex of modules over a commutative algebra $A$, the free graded co-commutative coalgebra $\vee^\bullet_A V$ over $V$, is the coalgebra (in the sense of co-monoids in complexes of modules) whose underlying complex is the graded-symmetric tensor algebra $S^\bullet_A V$ over $A$, and who co-product is the co-concatenation product.