On tree coalgebras and coalgebra presentations

Abstract

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t . The universal property of this family of coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique A t which takes the root of t to s. Consequently, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems. In contrast, for transition systems expressed as coalgebras over the finite-power-set functor we show that there are systems which fail to be filtered colimits of finitely presentable (=finite) ones. Surprisingly, if λ is an uncountable cardinal, then λ-presentation is always well-behaved: whenever an endofunctor F preserves λ-filtered colimits (i.e., is λ-accessible), then λ-presentable coalgebras are precisely those whose underlying objects are λ-presentable. Consequently, every F coalgebra is a λ-filtered colimit of λ-presentable coalgebras; thus Coalg F is a locally λ-presentable category. (This holds for all endofunctors of λ-accessible categories with colimits of ω-chains.) Corollary: A set functor is bounded at λ in the sense of Kawahara and Mori iff it is λ +-accessible.