Adámek, Jiří; Borceux, Francis; Lack, Stephen; Rosický, Jiří A classification of accessible categories. (English) Zbl 1010.18005 J. Pure Appl. Algebra 175, No. 1-3, 7-30 (2002). Let \(\mathbb{D}\) be a collection of small categories then the authors say that a (small) category \(\mathcal C\) is \(\mathbb{D}\)-filtered if \(\mathcal C\)-colimits commute in \(\mathcal{S}et\) with \(\mathbb{D}\)-limits. If a category \(\mathcal K\) has \(\mathbb{D}\)-filtered colimits then an object \( A\) of \(\mathcal K\) is called \(\mathbb{D}\)-presentable if the hom-functor \(\mathcal K(A,-):\mathcal K \to \mathcal{S}et\) preserves \(\mathbb{D}\)-filtered colimits. The authors say that a collection \(\mathbb{D}\) is sound if any category \(\mathcal C\) is \(\mathbb{D}\)-filtered whenever the category of cocones for any functor \(S:\mathcal D^{op}\to \mathcal C\) with \(\mathcal D\in \mathbb{D}\) is connected. For a sound collection \(\mathbb{D}\) a characterization of \(\mathbb{D}\)-continuous functors into \(\mathcal{S}et\) is given. A category \(\mathcal K\) is said to be \(\mathbb{D}\)-accessible if \(\mathbb{D}\) is sound, \(\mathcal K\) has \(\mathbb{D}\)-filtered colimits and there is a small set \(\mathcal A\) of \(\mathbb{D}\)-presentable objects of \(\mathcal K\) such that every object of \( \mathcal K\) is a \(\mathbb{D}\)-filtered colimit of objects from \(\mathcal A\).The paper develops a theory of \(\mathbb{D}\)-accessible categories that refines a theory of accessible (and locally presentable) categories. Since any \(\mathbb{D}\)-accessible category is also accessible, \(\mathbb{D}\)-accessible categories give a classification of accessible categories. The relation between \(\mathbb{D}\)-accessible categories and sketches is discussed. It is proved that the free completion under colimits in the quasi-category \(\mathcal{C}at\) of all categories distributes over free completion under limits in \(\mathcal{C}at\). Many examples illustrating the new concept are given. Reviewer: Václav Koubek (Praha) Cited in 1 ReviewCited in 23 Documents MathOverflow Questions: \(\mathbb D\)-weighted flatness of functors MSC: 18C35 Accessible and locally presentable categories 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18A35 Categories admitting limits (complete categories), functors preserving limits, completions 18C30 Sketches and generalizations Keywords:accessible category; sketch; locally presentable category; filtered colimit; free completion PDFBibTeX XMLCite \textit{J. Adámek} et al., J. Pure Appl. Algebra 175, No. 1--3, 7--30 (2002; Zbl 1010.18005) Full Text: DOI References: [1] J. Adàmek, F.W. Lawvere, J. 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