From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8769 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Indiscrete objects in a functor category Date: Sat, 19 Dec 2015 11:48:21 +0100 Message-ID: Reply-To: Aleks Kissinger NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1450569763 25843 80.91.229.3 (20 Dec 2015 00:02:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 20 Dec 2015 00:02:43 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sun Dec 20 01:02:36 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 1aARSN-0003qx-DF for gsmc-categories@m.gmane.org; Sun, 20 Dec 2015 01:02:35 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34011) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1aARRH-00081s-TC; Sat, 19 Dec 2015 20:01:27 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1aARRG-00021K-F2 for categories-list@mlist.mta.ca; Sat, 19 Dec 2015 20:01:26 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8769 Archived-At: 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/ ]