categories: technical question about omega-categories
Philippe Gaucher
gaucher@Mathematik.Uni-Marburg.de
Thu, 05 Jun 2003 15:47:42 +0100
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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.
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