Dear Steve, With regard to your query, I reproduce a message sent by Peter Johnstone to this list in March 2007 providing sufficient (and possibly necessary?) conditions for Lan_E to be comonadic. "A further attempt to provide a general context for Richard's observation: let f: C --> D be a functor between small categories having a right multi-adjoint in the sense of Diers, i.e. such that, for each object b of D, the comma category (f \downarrow b) is a disjoint union of categories with terminal objects. (Note that this is always the case when C is discrete, as in the example considered by Richard, since then the (f \downarrow b) are also discrete.) Then the left Kan extension functor f_!: [C,Set] --> [D,Set] can be constructed using only coproducts rather than more general colimits, from which it follows easily that it is faithful and preserves equalizers. Hence it is comonadic. (I suspect that this may be a necessary as well as a sufficient condition for comonadicity of f_!, but I don't yet have a proof.)" With best wishes, Richard --On 12 December 2008 12:57 Steve Lack wrote:
Does anyone know anything about when functors E:C-->D have the property that the functor Lan_E:[C,Set]-->[D,Set] given by left Kan extension is comonadic?
Actually I'm interested in something a bit more general. Let E:C-->D be a functor. I'm happy to suppose that it is bijective on objects and faithful. Then Lan_E has a right adjoint, given by restriction along E. Let W be the induced comonad on [D,Set], and [D,Set]^W its category of coalgebras. The comparison K:[C,Set]-->[D,Set]^W has a right adjoint R, constructed using equalizers in [C,Set]. Comonadicity would mean that this adjunction is an equivalence; what I really want to know is when/whether R is fully faithful, so that K is a reflection onto a full subcategory. (This is equivalent to Lan_E preserving the equalizers which are used to construct R.)
This seems like something topos theorists might know about.
Steve.