We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B∞-algebra. We construct a universal enveloping functor U2 from B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form...

The coalgebraic view on classes and objects is elaborated to include inheritance. Inheritance in coalgebraic speciication (of classes) will be understood dually to parametrization in algebraic speciication. That is, inheritance involves restriction (specialization), where parametrization involves extension. And cofree constructions are \best" restrictions, like free constructions are \best" ext...

This paper explicitly constructs cofree coalgebras over operads in the category of DG-modules. It is shown that the existence of an operadaction on a coalgebra implies a “generalized coassociativity” that facilitates the construction. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). In the pointed irreducible case, this constructio...

The non-commutative sequoid operator on games was introduced to capture algebraically the presence of state in history-sensitive strategies in game semantics, by imposing a causality relation on the tensor product of games. Coalgebras for the functor A — i.e., morphisms from S to A S — may be viewed as state transformers: if A has a final coalgebra, !A, then the anamorphism of such a state tran...

Every variable-free logic program induces a PfPf -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variable-free logic program with a PfPf -coalgebra on Set and showed that, if C(PfPf ) is the cofree comonad on ...

We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras to model the exponential is a model of differential linear logic. Thus, in this semantics, proof denotations have natural derivatives. We give several examples of these derivatives.