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Jeremy Gibbons (Mar 13 2025 at 10:02):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)?
Ivan Di Liberti (Mar 13 2025 at 13:27):See Locally Presentable and Accessible categories Thm 1.5, its immediate corollary, Subsection 1.6 and finally Corollary 1.7.
Jeremy Gibbons (Mar 13 2025 at 14:28):Aha - perfect! Thank you.
Josselin Poiret (Mar 13 2025 at 15:03):the stream functor is basically , and notice that maps don't necessarily factor through one of the , which is what co-continuity would entail.
Vincent Moreau (Mar 13 2025 at 15:28):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.
Jeremy Gibbons (Mar 13 2025 at 18:43):Thanks, both. The Stream example is helpful to me, but I'm afraid I'm no topologist...