From mboxrd@z Thu Jan  1 00:00:00 1970
X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1835
Path: news.gmane.org!not-for-mail
From: Philippe Gaucher <gaucher@irmast2.u-strasbg.fr>
Newsgroups: gmane.science.mathematics.categories
Subject: technical question about omega-categories
Date: Fri, 9 Feb 2001 19:36:33 +0100 (MET)
Message-ID: <200102091836.TAA07095@irmast2.u-strasbg.fr>
Reply-To: Philippe Gaucher <gaucher@irmast2.u-strasbg.fr>
NNTP-Posting-Host: main.gmane.org
Mime-Version: 1.0
Content-Type: TEXT/plain; charset=us-ascii
X-Trace: ger.gmane.org 1241018135 704 80.91.229.2 (29 Apr 2009 15:15:35 GMT)
X-Complaints-To: usenet@ger.gmane.org
NNTP-Posting-Date: Wed, 29 Apr 2009 15:15:35 +0000 (UTC)
To: categories@mta.ca
Original-X-From: rrosebru@mta.ca Fri Feb  9 15:18:16 2001 -0400
Return-Path: <cat-dist@mta.ca>
Original-Received: (from Majordom@localhost)
	by mailserv.mta.ca (8.11.1/8.11.1) id f19IdwF30454
	for categories-list; Fri, 9 Feb 2001 14:39:58 -0400 (AST)
X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f
Content-MD5: shbGAGcLjrmlBTk82w0Syw==
X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.3.5 SunOS 5.7 sun4u sparc 
Original-Sender: cat-dist@mta.ca
Precedence: bulk
X-Keywords: 
X-UID: 17
Original-Lines: 44
Xref: news.gmane.org gmane.science.mathematics.categories:1835
Archived-At: <http://permalink.gmane.org/gmane.science.mathematics.categories/1835>

Dear all,


Let C be an omega-category (strict, globular).

Let U be the forgetful functor from strict globular omega-categories
to globular sets. And let F be its left adjoint.

Let us suppose that we are considering an equivalence relation R on UC
(the underlying globular set of C) such that the source and target
maps pass to the quotient : i.e. one can deal with the quotient globular
set UC/R.

The canonical morphism of globular sets UC --> UC/R induces a morphism
of omega-categories F(UC) --> F(UC/R) by functoriality of F.

Consider the following push-out in the category of omega-categories :


F(UC) ----->  F(UC/R)
  |               |
  |               |
  |               |
  v               v
  C   --------->  D


The morphism F(UC)-->C (the counit of the adjonction) is surjective on
the underlying sets.

The morphism F(UC)-->F(UC/R) is generally not surjective on the
underlying sets : because by taking the quotient by R, one may add
composites in F(UC/R) which do not exist in F(UC).

However the intuition tells (me) that the morphism F(UC/R)-->D is
surjective on the underlying sets : this morphism only adds in F(UC/R)
the calculation rules of C : this is precisely what I want by
introducing D. But I cannot see why with a rigorous mathematical
argument.


Thanks in advance. pg.