Properties of co-operations: diagrammatic proofs

Traditionally, terms together with Sweedler notation are used to express computations in (generalized) bialgebras. Here, an algebra is a vector space equipped with an operation μ : ⊗→  and a bialgebra is an algebra  equipped with a co-operation δ :  →  ⊗ . The operation μ must be (bi)linear and satisfy some properties, for instance associativity and/or commutativity. Similarly, the co-operation δ must be linear and satisfy some co-properties, for instance coassociativity and/or co-commutativity. Furthermore, a compatibility relation between μ and δ must be satisfied for instance the Hopf identity. For more details, see (Loday 2008).