further on Garner's question

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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.)

Incidentally, note that if C and D are any two categories with
the same set of objects, we can construct a comonadic
adjunction (in either direction) between [C,Set] and [D,Set],
by `interposing' the discrete category with the same objects, in the
manner of Richard's example. Indeed, we don't even need the
categories to have the same objects: all we need is to find a
set which surjects onto both ob C and ob D.

Peter Johnstone

On Sat, 10 Mar 2007, Prof. Peter Johnstone wrote:

> Here's at least a partial conceptual explanation of
> Richard Garner's "curiosity". There are really three
> categories involved, all of them toposes: they are
> the functor categories [2,Set] where 2 denotes the
> discrete two-object category, [I,Set] where I denotes the
> category (* --> *) and [G,Set] where G has two objects and
> two parallel arrows between them. The inclusions
> f: 2 --> I and g: 2 --> G of course induce essential
> geometric morphisms (strings of three adjoint functors)
>
>           f           g
> [I,Set] <--- [2,Set] ---> [G,Set]
>
> and Richard's functors are simply the composites
> g_*f^* and f_!g^*. So it's no surprise that they
> should be adjoint: also, the adjunction (g^* -| g_*)
> is comonadic, because g is surjective on objects.
> The only oddity is that (f_! -| f^*) is also
> comonadic (it's obviously monadic, again because f
> is surjective on objects). As far as I can see, this is
> just an isolated fact: it isn't a particular case of any
> general result that I know.
>
> Peter Johnstone
>
> On Fri, 9 Mar 2007, Richard Garner wrote:
>
> >
> > While we're on the topic of directed graphs, can
> > anyone provide a satisfactory conceptual
> > explanation for the following curiosity?
> >
> > Let Ar(Set) be the arrow category of Set, and let
> > DGph be directed multigraphs, i.e., presheaves over
> > the parallel pair category as per Thomas' message.
> >
> > Prop: DGph is comonadic over Ar(Set)
> >
> > Proof: We have an adjunction U -| C as follows.
> >
> > U: DGph -> Ar(Set) sends a directed graph
> > s, t : A -> V to the coproduct injection V -> V + A.
> >
> > C: Ar(Set) -> DGph sends an arrow f : X -> Y to the
> > directed graph \pi_1, \pi_2 : X*X*Y -> X.
> >
> > It's easy to check that this is an adjunction, and
> > so we induce a comonad T = UC on Ar(Set), the
> > functor part of which sends f: X --> Y to the
> > coproduct injection X --> X + X*X*Y. Thus a
> > coalgebra structure f --> Tf consists of specifying
> > a map p: Y --> X + X*X*Y satisfying three axioms.
> >
> > These axioms force f: X --> Y to be an injection,
> > and the map p to be defined by cases: those y in Y
> > which lie in the image of f are sent to f^-1(y) in
> > the left-hand copy of X, whilst those y in Y that
> > are not in X are sent to some element (s(y), t(y),
> > y) of X*X*Y. Thus giving a T-coalgebra structure on
> > f:X --> Y is equivalent to giving a directed graph
> > structure s, t : Y \setminus f(X) --> X: and this
> > assignation extends to a functor T-Coalg --> DGph
> > which together with the canonical comparison
> > functor DGph --> T-Coalg gives us an equivalence of
> > categories, Q.E.D.
> >
> > --
> >
> > Richard Garner
>