Coalgebra

Definition
A coalgebra is the following data:

$$K$$ a field and $$C$$ be a $$K$$-vector space, equipped with the following data:


 * A linear map $$\Delta: C \to C \otimes_K C$$
 * A linear map $$\epsilon: C \to K$$

satisfying the following compatibility conditions:


 * Coassociativity:

$$(id_C \otimes \Delta) \circ \Delta = (\Delta \otimes id_C) \circ \Delta$$


 * Counit:

$$(id_C \otimes \epsilon) \circ \Delta = id_C = (\epsilon \otimes id_C) \circ \Delta$$