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A remark on accessible and axiomatizable categories. (English) Zbl 0849.18005
A category $${\mathcal A}$$ is accessible if there exists some regular cardinal $$\lambda$$ such that $${\mathcal A}$$ has $$\lambda$$-directed colimits and a set of $$\lambda$$-presentable objects whose closure under $$\lambda$$-directed colimits is $${\mathcal A}$$. A category is axiomatizable if it is equivalent to the category of models of some many-sorted infinitary first order theory. A category is bounded if it has a small dense subcategory. The paper proves that, under the large cardinal Vopěnka’s principle, a category with equalizers is accessible if and only if it is axiomatizable if and only if it is bounded.
##### MSC:
 18B99 Special categories 03C99 Model theory
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