nLab
free graded co-commutative coalgebra

Definition

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