Cofree coalgebra

In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space over a field exists, though it is more complicated than one might expect by analogy with the free algebra.

Definition

If V is a vector space over a field k, then the cofree coalgebra C(V) of V is a coalgebra together with a linear map C(V)→V, such that any linear map from a coalgebra X to V factors through a coalgebra homomorphism from X to C(V). In other words the functor C is right adjoint to the forgetful functor from coalgebras to vector spaces.

The cofree coalgebra of a vector space always exists, and is unique up to canonical isomorphism.

Cofree cocommutative coalgebras are defined in a similar way, and can be constructed as the largest cocommutative coalgebra in the cofree coalgebra.

Construction

References

• Block, Richard E.; Leroux, Pierre (1985), "Generalized dual coalgebras of algebras, with applications to cofree coalgebras", Journal of Pure and Applied Algebra 36 (1): 15–21, doi:10.1016/0022-4049(85)90060-X, ISSN 0022-4049, MR782637, http://dx.doi.org/10.1016/0022-4049(85)90060-X 
• Hazewinkel, Michiel (2003), "Cofree coalgebras and multivariable recursiveness", Journal of Pure and Applied Algebra 183 (1): 61–103, doi:10.1016/S0022-4049(03)00013-6, ISSN 0022-4049, MR1992043, http://dx.doi.org/10.1016/S0022-4049(03)00013-6