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