Sweedler notation
Comultiplication turns one element into a sum of tensor pairs, but writing out the sum with explicit indices, Δ(c) = Σ_i a_i ⊗ b_i, quickly drowns any calculation in subscripts. Sweedler notation is a deliberate, careful abuse of notation that suppresses the summation index and the dummy basis, letting you reason about coalgebras the way you reason about ordinary products.
The convention writes Δ(c) = Σ c(1) ⊗ c(2), or even just c(1) ⊗ c(2) with the sum understood, where the subscripts (1) and (2) are not specific elements but placeholders for the two tensor legs. Coassociativity makes the iterated coproduct unambiguous, so one writes (Δ ⊗ id)Δ(c) = (id ⊗ Δ)Δ(c) = Σ c(1) ⊗ c(2) ⊗ c(3) with three legs, and so on for higher coproducts.
The notation makes the Hopf axioms readable as near-arithmetic identities. The counit axiom becomes Σ ε(c(1)) c(2) = c = Σ c(1) ε(c(2)); the antipode axiom becomes Σ S(c(1)) c(2) = ε(c) 1 = Σ c(1) S(c(2)); a map φ is a coalgebra morphism iff Σ φ(c)(1) ⊗ φ(c)(2) = Σ φ(c(1)) ⊗ φ(c(2)). One must remember the hidden summation is real — the legs are not single elements — but with that caveat the notation is indispensable.
Introduced by Moss Sweedler in his 1969 book Hopf Algebras, and sometimes called sigma notation. A common variant writes Δ(c) = c' ⊗ c''; another drops the explicit Σ entirely. The cardinal sin is to treat c(1) and c(2) as if they were single, separable elements — they are entangled across the summation.