Subject: categories: technical question about omega-categories

From: Philippe Gaucher <[email-masked]>
Date: Fri, 9 Feb 2001 19:36:33 +0100 (MET)

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.