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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.
18B99 Special categories
03C99 Model theory
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