Multiplication, drawn as a map
Start where you are comfortable. An associative algebra over a field k is a vector space A together with a k-linear multiplication. The trick that makes everything dual-izable is to record that multiplication not as a binary operation a·b but as a single linear map m: A⊗A → A out of the tensor product — this is exactly what the universal property of the tensor product lets us do, since multiplication is bilinear. The unit element 1 becomes a map u: k → A sending the scalar 1 to 1∈A. Associativity and the unit law are now commuting diagrams of linear maps, not equations among elements.
Associativity says the two ways of bracketing a triple product agree: m∘(m⊗id) = m∘(id⊗m) as maps A⊗A⊗A → A. The unit law says m∘(u⊗id) = id = m∘(id⊗u) after identifying k⊗A ≅ A ≅ A⊗k. Nothing new yet — but writing it this way means every symbol can be reflected.
Flip every arrow
A coalgebra is what you get by reversing m and u. Comultiplication is a linear map Δ: C → C⊗C; the counit is ε: C → k. The two axioms are the mirror images of the algebra axioms. Coassociativity: (Δ⊗id)∘Δ = (id⊗Δ)∘Δ as maps C → C⊗C⊗C. Counit law: (ε⊗id)∘Δ = id = (id⊗ε)∘Δ. Read literally, the comultiplication takes one element and disperses it into a sum of tensor pairs, and the counit is a way to collapse a tensor factor back to a scalar.
The two examples to keep
First, the group algebra k[G] of a finite group G. As an algebra it is the familiar group algebra. As a coalgebra, define Δ(g) = g⊗g and ε(g) = 1 on each basis element g∈G, extended linearly. Coassociativity is immediate: both sides give g⊗g⊗g. Such a g is called a grouplike element — it co-multiplies into a perfect copy of itself. Second, a function-space coalgebra: the linear dual of a finite-dimensional algebra is automatically a coalgebra, because dualizing m: A⊗A → A gives a map A* → (A⊗A)* ≅ A*⊗A*. The dual space is where the two structures trade places.
Coassociativity check for the group algebra k[G].
Basis: the elements g in G. Define on basis vectors
Delta(g) = g (x) g, eps(g) = 1,
and extend k-linearly.
Left branch: (Delta (x) id) Delta(g)
= (Delta (x) id)(g (x) g)
= Delta(g) (x) g
= (g (x) g) (x) g
= g (x) g (x) g.
Right branch: (id (x) Delta) Delta(g)
= (id (x) Delta)(g (x) g)
= g (x) Delta(g)
= g (x) (g (x) g)
= g (x) g (x) g.
Equal. So coassociativity holds on a basis, hence everywhere.
Counit: (eps (x) id) Delta(g) = eps(g) g = 1.g = g. OK.
(id (x) eps) Delta(g) = g . eps(g) = g. OK.
Contrast: a *primitive* element x would have
Delta(x) = x (x) 1 + 1 (x) x, eps(x) = 0,
the coalgebra shadow of the Leibniz rule. We meet these in Guide 3.