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.
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Richard Garner