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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 –:
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