A classification of accessible categories
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.
scientific article; zbMATH DE number 1839059
scientific article