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Smyth and Plotkin (1979,1982) and Wand (1979) explored the existence of the initial algebra of a functor F. It basically boils down to requiring "co-continuity" of F: preservation of colimits of right chains.
Taylor (1999), Capretta et al (2005), Adamek et al (2007) discussed recursive coalgebras for a functor F: F-coalgebras c such that for any F-algebra a, the equation h = a . Fh . c has a unique solution in h. There's an important result identifying three separate sufficient conditions for c to be a recursive coalgebra (informally, variant functions, finite unfoldings, coalgebra homomorphism to the inverse of the initial algebra), stating that these coincide when F is a "finitary" functor, ie F preserves directed colimits, sometimes stated as "preserving filtered colimits".
Are these two things - preservation of colimits of right chains, and preservation of directed/filtered colimits - the same? If so, is there a good reason to give them different names, or is that just some historical accident? If not, why not? Counterexamples in SET-like categories are most helpful for me (since I'm using these notions in the context of types and programs).
I've always felt comfortable with the relevance of co-continuity; but now I come to reflect on it, I don't think I have a good grasp of why a functor (in particular, an endofunctor on SET) might fail to be co-continuous. Knowing that co-continuity is the same as finitariness might help. Is the Stream endofunctor on SET (taking A to infinite sequences of As) non-finitary, and non-co-continuous? What is the difference between F(colim<X_i>) and colim<F(X_i)>, in particular when F=Stream and X_i=F^i(0)?
See Locally Presentable and Accessible categories Thm 1.5, its immediate corollary, Subsection 1.6 and finally Corollary 1.7.
Aha - perfect! Thank you.
the stream functor is basically , and notice that maps don't necessarily factor through one of the , which is what co-continuity would entail.
Another example of non-cocontinuous functor is given by the ultrafilter monad on Set, whose algebras are compact Hausdorff spaces. It is the identity on finite sets. The map colim<F(X_i)> → F(colim<X_i>) for X_i the collection of finite subsets of some set X can be understood as the unit of this monad, which sends an element on its principal ultrafilter.
Thanks, both. The Stream example is helpful to me, but I'm afraid I'm no topologist...