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MORE ON INJECTIVITY IN LOCALLY PRESENTABLE CATEGORIES
 

Summary: MORE ON INJECTIVITY IN LOCALLY
PRESENTABLE CATEGORIES
J. Rosick' y 1) 2) , J. Ad' amek 1) and F. Borceux
Dedicated to Horst Herrlich on the occasion of his 60th birthday.
Abstract. Injectivity w.r.t. morphisms having –­presentable domains and codo­
mains is characterized: such injectivity classes are precisely those closed under prod­
ucts, –­directed colimits, and –­pure subobjects. This sharpens the result of the first
two authors (Trans. Amer. Math. soc. 336 (1993), 785­804). In contrast, for geometric
logic an example is found of a class closed under directed colimits and pure subob­
jects, but not axiomatizable by a geometric theory. A more technical characterization
of axiomatizable classes in geometric logic is presented.
I. Introduction
In [AR 1 ], classes of objects injective w.r.t. a set M of morphisms of a locally
presentable category K were characterized: they are precisely the classes closed
under products, –­directed colimits and –­pure subobjects for some cardinal – (see
Part II below for the concept of –­pure subobject). In fact, the formulation in [AR 1 ]
did not use –­pure subobjects, but accessibility of the class in question. However, a
full subcategory of K, closed under –­directed colimits, is accessible iff it is closed
under – 0 ­pure subobjects for some – 0 (see [AR 2 ], Corollary 2.36). The main result
of our paper is a ``sharpening'' of the previous result to a given regular cardinal –:

  

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

 

Collections: Computer Technologies and Information Sciences