Re: further on Garner's question

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

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.
>>>
>>> --
>>>
>>> Richard Garner
>>
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