# coalgebra

A coalgebra is a vector space $A$ over a field $\mathbb{K}$ with a $\mathbb{K}$-linear map $\Delta\colon A\to A\otimes A$, called the comultiplication, and a (non-zero) $\mathbb{K}$-linear map $\varepsilon\colon A\to\mathbb{K}$, called the counit, such that

 $\displaystyle(\Delta\otimes\mathrm{id})\circ\Delta$ $\displaystyle=$ $\displaystyle(\mathrm{id}\otimes\Delta)\circ\Delta\quad\mbox{(coassociativity)},$ $\displaystyle(\varepsilon\otimes\mathrm{id})\circ\Delta$ $\displaystyle=\mathrm{id}=$ $\displaystyle(\mathrm{id}\otimes\varepsilon)\circ\Delta.$