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.

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.

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

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

In addition to the graded algebra structure given by the tensor product isomorphisms T^jV ⊗ T^kV → T^j+kV for j, k ∈ N, T(V) has a graded coalgebra structure Δ : T(V) → T(V) ⊠ T(V) defined by extending

by linearity to all of T(V).

Here, the tensor product symbol ⊠ is used to indicate the tensor product used to define a coalgebra; it must not be confused with the tensor product ⊗, which is used to define the bilinear multiplication operator of the tensor algebra. The two act in different spaces, on different objects. Additional discussion of this point can be found in the tensor algebra article.

The sum above makes use of a short-hand trick, defining ${\displaystyle v_{0}=1\in \mathbb {F} }$ to be the unit in the field ${\displaystyle \mathbb {F} }$. For example, this short-hand trick gives, for the case of ${\displaystyle k=1}$ in the above sum, the result that

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