Associative Algebra - Coalgebras

Coalgebras

An associative unital algebra over K is given by a K-vector space A endowed with a bilinear map A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. If the bilinear map A×A→A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A⊗A→A (by the universal property of the tensor product), then we can view an associative unital algebra over K as a K-vector space A endowed with two morphisms (one of the form A⊗A→A and one of the form K→A) satisfying certain conditions which boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra.

There is also an abstract notion of F-coalgebra. This is vaguely related to the notion of coalgebra discussed above.