We extend our work in [1] to the case of Hopf comodule coalgebras. We introduce the cocylindrical module C♮H, where H is a Hopf algebra with bijective antipode and C is a Hopf comodule coalgebra over H. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra C>◭H and ∆(C♮H), the cocyclic module related to the diagonal of C♮H. We approximate HC(C >...

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Set-monad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, t...

We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p...

A propositional logic program P may be identified with a PfPf -coalgebra on the set of atomic propositions in the program. The corresponding C(PfPf )-coalgebra, where C(PfPf ) is the cofree comonad on PfPf , describes derivations by resolution. Using lax semantics, that correspondence may be extended to a class of first-order logic programs without existential variables. The resulting extension...

We extend the 2-representation theory of finitary 2-categories to certain with infinitely many objects, denoted locally 2-categories, and classical classification results simple transitive 2-representations weakly fiat this environment. also consider a grading, prove associated coalgebra 1-morphisms have homogeneous structure. use these classify classes cyclotomic 2-Kac-Moody algebras.

We introduce and study bimeasurings from pairs of bialge-bras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler's universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant functor on the category of bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in the category o...

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...