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Cofree coalgebra

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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 F, 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

C(V) may be constructed as a completion of the tensor coalgebra T(V) of V. For kN = {0, 1, 2, ...}, let TkV denote the k-fold tensor power of V:

with T0V = F, and T1V = V. Then T(V) is the direct sum of all TkV:

In addition to the graded algebra structure given by the tensor product isomorphisms TjVTkVTj+kV for j, kN, T(V) has a graded coalgebra structure Δ : T(V) → T(V) ⊗ T(V) defined by extending

by linearity to all of T(V). This coproduct does not make T(V) into a bialgebra, but is instead dual to the algebra structure on T(V), where V denotes the dual vector space of linear maps VF. Here an element of T(V) defines a linear form on T(V) using the nondegenerate pairings

induced by evaluation, and the duality between the coproduct on T(V) and the product on T(V) means that

This duality extends to a nondegenerate pairing

where

is the direct product of the tensor powers of V. (The direct sum T(V) is the subspace of the direct product for which only finitely many components are nonzero.) However, the coproduct Δ on T(V) only extends to a linear map

with values in the completed tensor product, which in this case is

and contains the tensor product as a proper subspace:

The completed tensor coalgebra C(V) is the largest subspace C satisfying

which exists because if C1 and C2 satisfiy these conditions, then so does their sum C1 + C2.

It turns out[1] that C(V) is the subspace of all representative elements:

Furthermore, by the finiteness principle for coalgebras, any fC(V) must belong to a finite dimensional subcoalgebra of C(V). Using the duality pairing with T(V), it follows that fC(V) if and only if the kernel of f on T(V) contains a two-sided ideal of finite codimension. Equivalently,

is the union of annihilators I0 of finite codimension ideals I in T(V), which are isomorphic to the duals of the finite dimensional algebra quotients T(V)/I.

References

  1. ^ Hazewinkel 2003
  • 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, MR 0782637
  • 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, MR 1992043
  • cofree coalgebra at the nLab