comonadicity of Lan_E

comonadicity of Lan_E

[Steve Lack]

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.

Re: comonadicity of Lan_E

[Richard Garner]

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.
>
>
>
>