Re: Garner's question

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

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