Codex @codex  0

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

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.

A right comodule over a coalgebra $(C,\Delta ,\epsilon )$ over a commutative ring $k$ is a $k$-module $M$ with a coaction $\rho :M\to M{\otimes }_{k}C$ satisfyingusing the canonical tensor identifications. In Sweedler notation, $\rho (m)=\sum m_{(0)}\otimes m_{(1)}$. The equations are the reversed-arrow counterparts of associativity and the unit law for an algebra module.^[ref]

For the group algebra coalgebra $kG$ with $\Delta (g)=g\otimes g$, a direct sum $M={⨁}_{g\in G}{M}_g$ gives a right comodule by $\rho (m_g)=m_g\otimes g$. Conversely, expanding a coaction in the basis of group elements recovers this direct sum decomposition. Thus comodules over this coalgebra encode a decomposition indexed by $G$, whereas modules over the same group algebra encode a group representation. More generally, the definition extends to comonoids in a monoidal category whenever the relevant tensor action is specified.

The structure map $\Delta :C\to C\otimes C$ of a comonoid; it is coassociative up to the ambient associator.

The structure map $\epsilon :C\to I$ of a comonoid, with $(\epsilon \otimes 1)\Delta$ and $(1\otimes \epsilon )\Delta$ equal to the inverse unitors. This is distinct from the counit of an adjunction.