From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8771 Path: news.gmane.org!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: Indiscrete objects in a functor category Date: Sun, 20 Dec 2015 15:25:38 +0000 (GMT) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1450644667 32629 80.91.229.3 (20 Dec 2015 20:51:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 20 Dec 2015 20:51:07 +0000 (UTC) Cc: categories To: Aleks Kissinger Original-X-From: majordomo@mlist.mta.ca Sun Dec 20 21:51:00 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 1aAkwU-0003wI-Ot for gsmc-categories@m.gmane.org; Sun, 20 Dec 2015 21:50:58 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34127) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1aAkw0-0001Cj-BT; Sun, 20 Dec 2015 16:50:28 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1aAkvy-0005N8-HC for categories-list@mlist.mta.ca; Sun, 20 Dec 2015 16:50:26 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8771 Archived-At: The forgetful functor [C,Set] --> Set/ob C always has a right adjoint, given by right Kan extension along the inclusion C_0 --> C, where C_0 is the discrete category with the same objects as C. For the same construction in a more general context, see B2.3.16 in `Sketches of an Elephant'. Peter Johnstone On Sat, 19 Dec 2015, Aleks Kissinger wrote: > It's common to describe the category of (directed, multi-) graphs as a > functor category Graph := [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 = pi2, s = pi2, t = 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/ ]