Terminal sequences for accessible endofunctors

Abstract

We consider the behaviour of the terminal sequence of an accessible endofunctor T on a locally presentable category K. The preservation of monics by T is sufficient to imply convergence, necessarily to a terminal coalgebra. We can say much more if K is Set, and κ is ω. In that case it is well known that we do not necessarily get convergence at ω, however we show that to ensure convergence we don't need to go to a higher cardinal, just to the next limit ordinal, ω + ω.For an ω-accessible endofunctor T on Set the construction of the terminal coalgebra can thus be seen as a two stage construction, with each stage being finitary. The first stage obtains the Cauchy completion of the initial T-algebra as the ω-th object in the terminal sequence Aω. In the second stage this object is pruned to get the final coalgebra Aω+ω. We give an example where Aω is the solution of the corresponding domain equation in the category of complete ultra-metric spaces.