Coalgebra

Remarks

Example: for a set $S$, let $C = K[S]$ be the vector space with basis $S$, and put $\Delta(s) = s \otimes s$ and $\varepsilon(s) = 1$ for $s \in S$. Both sides of clause 1 send $s$ to $s \otimes s \otimes s$, and both maps in clause 2 send $s$ to $1 \cdot s = s$. Non-example: the same $\Delta$ with $\varepsilon(s) = 0$ for all $s$ fails clause 2, since $(\varepsilon \otimes \id)\Delta(s) = 0 \ne s$. Another example is $K[x]$ with $\Delta(x^n) = \sum_{k=0}^{n} \binom{n}{k} x^k \otimes x^{n-k}$ and $\varepsilon(x^n) = 1$ if $n = 0$ and $0$ otherwise. The definition works in the same way over a commutative ring in place of $K$. Over a field, every element of a coalgebra lies in a finite-dimensional subcoalgebra.