From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3693 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Garner's question Date: Sat, 10 Mar 2007 11:49:41 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019465 9854 80.91.229.2 (29 Apr 2009 15:37:45 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:45 +0000 (UTC) To: Categories mailing list Original-X-From: rrosebru@mta.ca Sat Mar 10 14:34:44 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 10 Mar 2007 14:34:44 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HQ6JJ-0004s6-7A for categories-list@mta.ca; Sat, 10 Mar 2007 14:28:53 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 47 Original-Lines: 71 Xref: news.gmane.org gmane.science.mathematics.categories:3693 Archived-At: 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