An algebraic deformation theory of coalgebra morphisms is constructed.

The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [7] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [5], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra [3] can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras A = {A α } α∈π ...

rule induction. Together, an abstract rule of this form and a U-structured B-coalgebra, (X ,h : UX → BUX ), provide parameterised recursion data (C ,Σ,T,UX ,BU T̃X , fρ,X , gX ,h) where the morphisms fρ,X , gX ,h are defined as follows. fρ,X = Σ(TUX × BU T̃X ) Σ(uX×... ) −−−−−→ Σ(UT̃X × BU T̃X ) ρ T̃ X −→ BU T̃ T̃ X BUμ̃X −→ BU T̃X gX ,h = UX h −→ BUX BUη̃X −−−→ BU T̃X . (6.2.12) Hence an abstract rule ρ ...

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra α : X → FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor F in the category PMet of pseudometric spaces. We present two differen...

Final coalgebras capture system behaviours such as streams, infinite trees and processes. Algebraic operations on elements of a final coalgebra can be defined by distributive laws (of a syntax Σ functor over a behaviour functor F ). Such distributive laws correspond to abstract specification formats. One such format is a generalization of the GSOS rules known from structural operational semanti...

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the final coalgebra. Inspired by Ésik and Maletti’s notion of proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rati...

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843– 2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coal...

In [17], Esakia uses the Vietoris topology to give a coalgebra-flavored definition of topological Kripke frames, thus relating the Vietoris topology, modal logic and coalgebra. In this chapter, we sketch some of the thematically related mathematical developments that followed. Specifically, we look at Stone duality for the Vietoris hyperspace and the Vietoris powerlocale, and at recent work com...

A cotensor product A HP of an H-Hopf Galois extension A and a C-coalgebra Galois extension P , such that P is an (H,C)-bicomodule, is analyzed. Conditions are stated, when A HP is a C-coalgebra Galois extension and when there exists a strong connection on A HP . Two examples are given, in both, A and P are Matsumoto spheres, and H = C = C(U(1)).