Co-algebra

A module $A$ over a commutative ring $k$ with two homomorphisms $\phi$ and $\epsilon$ such that the diagrams $$ \begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ \phi\,\downarrow & \ & \downarrow\,1 \otimes \phi \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array} $$ and

are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring .

Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. Hopf algebra).

References

[1]	S. MacLane, "Homology" , Springer (1963)

Comments

Given a co-algebra over . Let be the module of -module homomorphisms from to . For define the product by the formula , where is identified with . For any two -modules define by . Then the multiplication on can also be seen as the composite . The element is a unit element for this multiplication making an associative algebra with unit, the dual algebra. In general the mapping is not an isomorphism and there is no natural -module homomorphism . Thus there is no equally natural construction associating a co-algebra to an algebra over , even when is a field. In that case there does however exist an adjoint functor to the functor which associates to a co-algebra its dual algebra, i.e. for , , where and denote, respectively, the category of -algebras and the category of -co-algebras, [a2]; cf. also Hopf algebra. But if is free of finite rank over then is an isomorphism and the dual co-algebra can be defined.

Let be the set . Let and define

Then is a co-algebra.

If and are two co-algebras, then a morphism of co-algebras is a -module morphism such that and . A co-ideal of a co-algebra is a -submodule such that and .

A co-module over a co-algebra is a -module with a -module morphism such that and the canonical isomorphism . There are obvious notions of homomorphisms of co-modules, etc.

References

[a1]	M. Sweedler, "Hopf algebras" , Benjamin (1969)
[a2]	E. Abe, "Hopf algebras" , Cambridge Univ. Press (1980)