Very interesting! I was aware that one could construct such comonadic adjunctions in a limited variety of other cases, but that one can do so for essentially any pair of presheaf categories is (at least to me) a little surprising. Of course, even in the case where C and D have the same objects, there is nothing "canonical" about the comonadic adjunction [C, Set] -| [D, Set] which we construct in the way Peter has indicated, since we have first to choose a bijection between the objects of C and the objects of D. Richard Garner --On 10 March 2007 15:58 Prof. Peter Johnstone wrote:
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.
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Richard Garner