From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3696 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: further on Garner's question Date: Mon, 12 Mar 2007 09:23:16 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019467 9873 80.91.229.2 (29 Apr 2009 15:37:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:47 +0000 (UTC) To: Categories mailing list Original-X-From: rrosebru@mta.ca Mon Mar 12 15:14:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 12 Mar 2007 15:14:43 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HQotf-0007Gx-6g for categories-list@mta.ca; Mon, 12 Mar 2007 15:05:23 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 50 Original-Lines: 122 Xref: news.gmane.org gmane.science.mathematics.categories:3696 Archived-At: 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 >> > > >