comultiplication

Multiplication fuses two inputs into one output. Comultiplication, often called the coproduct, runs the other way: it takes one element and produces a sum of tensor pairs, telling you all the ways the element can be 'pulled apart' into two pieces. For a function on a group, comultiplication encodes the group's multiplication: the value at a product gets expressed through values at the factors.

Precisely, a comultiplication on a vector space C is a linear map Δ : C -> C ⊗ C that is coassociative: (Δ ⊗ id) ∘ Δ = (id ⊗ Δ) ∘ Δ. Coassociativity is what lets you write iterated coproducts unambiguously, just as ordinary associativity lets you drop parentheses in products. Together with a compatible counit, a comultiplication makes C a coalgebra.

The motivating geometric picture: if G is a finite group and C = k[G] is the space of functions on G, then Δ(f) is the function on G × G given by Δ(f)(x, y) = f(xy), viewed inside k[G] ⊗ k[G] ≅ k[G × G]. So comultiplication is literally the dual, or pullback along multiplication, of the group law. Quantum groups arise by deforming exactly this map while keeping coassociativity.