| | |
Summary: On Pure Quotients and Pure Subobjects
J. Adamek and J. Rosicky y
May 17, 2004
Abstract
In the theory of accessible categories pure subobjects, i.e. ltered
colimits of split monomorphisms, play an important role. Here we
investigate pure quotients, i.e., ltered colimits of split epimorphisms.
For example, in abelian, nitely accessible categories, these are pre-
cisely the cokernels of pure subobjects, and pure subobjects are pre-
cisely the kernels of pure quotients.
1 Introduction
The concept of a -pure subobject stems from module and model theory, and
has been rst categorically formulated by S. Fakir [3]. In our monograph [1]
we have simpli ed that de nition, and have proved that -pure subobjects
play a central role in the theory of accessible categories of C. Lair [5] and
M. Makkai { R. Pare [6]. In particular the following results hold (recall that
\accessibly embedded" means full and closed under - ltered colimits for
some regular cardinal ):
(a) every accessible category K is closed in K under -pure subobjects for
some ,
|
