A classification of accessible categories

For a suitable collection image of small categories, we define the image-accessible categories, generalizing the λ-accessible categories of Lair, Makkai, and Paré; here the λ-accessible categories are seen as the image-accessible categories where image consists of the λ-small categories. A small category image is called image-filtered when image-colimits commute with image-limits in the category of sets. An object of a category is called image-presentable when the corresponding representable functor preserves image-filtered colimits. The image-accessible categories are then the categories with image-filtered colimits and a small set of image-presentable objects which is “dense with respect to image-filtered colimits”.