 Coalgebra

A coalgebra over a commutative ring $k$ is a module $C$ equipped with linear maps $\Delta :C\to C{\otimes }_{k}C$ and $\epsilon :C\to k$, called comultiplication and counit, satisfyingThe first identity is coassociativity: splitting into three tensor factors does not depend on which factor is split first. The second identity says that applying the counit to either output recovers the original element, under the canonical identifications $k\otimes C\cong C\cong C\otimes k$. Categorically, this is a comonoid in the monoidal category of $k$-modules.^[ref]

For a group algebra $kG$, define $\Delta (g)=g\otimes g$ and $\epsilon (g)=1$ on each group element, then extend linearly. The two axioms hold on this basis, hence everywhere. For a finite-dimensional associative algebra over a field, dualizing multiplication and the unit gives a coalgebra on its dual space. A coalgebra packages the splitting operation that is used by a comodule, rather than an operation multiplying two input elements.