Abstract:
Coalgebras are a category theoretic construction of interest to theoretical computer science, defined by an endofunctor T : C → C over a category C. A class K of T-coalgebras is a covariety if it is closed under coproducts, codomains of epi coalgebraic morphisms and subcoalgebras, and a behavioural covariety if it is also closed under images of bisimulations. Often the forgetful functor from the category of T-coalgebras to C has a right adjoint. This adjunction defines a comonad G T.
Goldblatt [10] investigated this situation over the category Set, introducing the notion of a pure subcomonad of a comonad, and showing a bijective correspondence between (equivalence classes of) pure subcomonads of GT and behavioural covarieties of T-coalgebras. This thesis demonstrates what restrictions need to be applied to an arbitrary category C and endofunctor T : C → C to attain this bijection. All restrictions on C and T are shown to apply to an endofunctor ()I on a category Set that is useful for modelling automata. The thesis goes on to show that pure subcomonads give rise to subcoalgebras of the final coalgebra called coequations over 1, and demonstrates a bijective correspondence between behavioural co-varieties of T-coalgebras and coequations over 1.
Finally the thesis demonstrates a bijective correspondence between covarieties of T-coalgebras and a class of subcomonads called regulating subcomonads.
