From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8770 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Re: Indiscrete objects in a functor category Date: Sun, 20 Dec 2015 09:48:12 +0100 Message-ID: References: <002432A4-ECC1-42DA-B9B8-9B2F8B42EF52@mq.edu.au> Reply-To: Aleks Kissinger NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1450644598 31510 80.91.229.3 (20 Dec 2015 20:49:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 20 Dec 2015 20:49:58 +0000 (UTC) Cc: categories list To: Steve Lack Original-X-From: majordomo@mlist.mta.ca Sun Dec 20 21:49:50 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1aAkvO-0001x8-Kb for gsmc-categories@m.gmane.org; Sun, 20 Dec 2015 21:49:50 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34122) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1aAkua-00015s-1y; Sun, 20 Dec 2015 16:49:00 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1aAkuY-0005M7-7u for categories-list@mlist.mta.ca; Sun, 20 Dec 2015 16:48:58 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8770 Archived-At: Yes, indiscrete isn't exactly right. Perhaps something like "saturated" is more appropriate. Right Kan extensions give a nice way to put this, which makes it clear that this extends beyond finite base category as well. So, the refinement of my question is: have people studied (left or) right Kan extensions over the inclusion into C of its associated discrete category C0? It seems to me for instance that taking C =3D the simplicial category and C0 =3D natural numbers would have been studied, e.g. in forming something like "coloured simplicial sets" as a slice. On 20 December 2015 at 03:28, Steve Lack wrote: > Dear Aleks, > > I would say that the known construction of which this is a special case i= s > right Kan extensions. The forgetful functor from graphs to pairs of sets = is > given by restriction along the inclusion in what you call 2 of the subcat= egory > with the same object but no non-identity arrows. Thus the right adjoint i= s given > by right Kan extension along this inclusion. Such right Kan extensions ca= n always > be constructed using limits; in your case, because the domain of the func= tor along > which you are extending is discrete, these limits are actually products. > > By the way, I would not use =E2=80=9Cindiscrete=E2=80=9D in this context.= For me indiscrete would > refer to things in the image of the right adjoint to the functor which as= sociates to a > graph its set of vertices. The set of edges of the indiscrete graph would= then be V x V. > > Regards, > > Steve Lack. > > > >> On 19 Dec 2015, at 9:48 PM, Aleks Kissinger wrote: >> >> It's common to describe the category of (directed, multi-) graphs as a >> functor category Graph :=3D [2, Set], where 2 here is the category with >> 2 objects and 2 parallel arrows (s & t). >> >> For a pair of sets (V,E), one can construct the indiscrete graph >> I(V,E) as a graph with vertices V and edges E x V x V, where the >> source and target maps are just the 2nd and 3rd projection >> respectively. This gives a right adjoint to the forgetful functor from >> Graph to pairs of sets. This enables one to construct a category of >> graphs with a fixed set of vetex/edge labels as a slice over Graph: >> >> Graph / I(Lv, Le) >> >> since a graph hm G --> I(Lv,Le) is the same as a map U(G) --> (Lv,Le), >> which is just a pair of functions assigning labels to the vertices and >> edges of G. >> >> >> This seems to me like a pretty standard trick, which extends to any >> functor category from a C which is in some sense "suitably acyclic". >> For instance, consider a category of "partitioned graphs" [3, Set], >> where 3 has objects (P,V,E) and arrows: >> >> E --s--> V, E --t--> V, and V --p--> P >> >> where, p assigns each of the vertices a partition. For a triple >> (P,V,E) we can form the indiscrete partitioned graph with: >> >> - partitions P >> - vertices V x P >> - edges E x (V x P) x (V x P) >> - p =3D pi2, s =3D pi2, t =3D pi3 >> >> which gives a right-adjoint to the forgetful functor from partitioned >> graphs to triples of sets. This is clearly an instance of a general >> recipe, whereby you start with the objects with no arrows out, and >> work your way backwards, always adding copies of the codomain of every >> out-arrow. Again one can attach labels to partitioned graphs by >> slicing: >> >> [3,Set] / I(Lp,Lv,Le) >> >> >> So, my question: Is the general case a known/studied construction? If >> so, could someone provide a reference? >> >> >> >> Best, >> >> Aleks >> >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]