Definition 3

A graded bigebra (resp. skew graded bigebra) over a ring A is a set E with a graded A-algebra structure of type $\mathbf{N}$ and a graded A-cogebra structure of type $\mathbf{N}$, with the same underlying graded A-module structure and such that:

(1) The A-algebra E is associative and unital.

(2) The A-cogebra E is coassociative and counital. (3) The coproduct c : E \to E \otimesA E is a homomorphism of the graded algebra E into the graded algebra E \otimes_A E* (resp. graded algebra E^g \otimesA E* (cf. § 4, no. 7)). (4) The counit $\gamma$ of E is a homomorphism of the graded algebra E into the algebra A (with the trivial graduation) such that, if e denotes the unit element of the A-algebra E, $\gamma(e) = 1$.

If E is a graded bigebra whose graduation is trivial, E is called simply a bigebra. A graded bigebra is called commutative (resp. cocommutative) if the underlying algebra is commutative (resp. if the underlying cogebra is cocommutative); a skew graded bigebra is called anticommutative (resp. anticocommutative) if the underlying graded algebra is anticommutative (resp. if the underlying graded cogebra is anticocommutative).

It follows from Definition 3 and no. 2, Proposition 5 that, for a graded bigebra or a skew graded bigebra E,

If E and F are two graded bigebras (resp. two skew graded bigebras), a mapping $\phi : E \to F$ is called a graded bigebra morphism (resp. skew graded bigebra morphism) if: (1) $\phi$ is a graded algebra morphism (and hence maps the unit element of E to the unit element of F); (2) $\phi$ is a graded cogebra morphism such that, if $\gamma$ and $\gamma'$ are the respective counits of E and F, then $\gamma = \gamma' \circ \phi$.