On tree coalgebras and coalgebra presentations

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 At. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique At which takes the root of t to s. Moreover, 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 CoalgF is a locally λ–presentable category. (This holds for all endofunctors of λ–accessible categories with colimits of ω-chains.) Corollary: λ–accessible set functors are bounded at λ in the sense of Kawahara and Mori and, conversely, boundedness at λ implies λ+–accessibility.