  We introduce the notion of a definable category--a category equivalent to a
full subcategory of a locally finitely presentable category that is closed
under products, directed colimits and pure subobjects. Definable subcategories
are precisely the finite-injectivity classes. We prove a $2$-duality between
the $2$-category of small exact categories and the $2$-category of definable
categories, and provide a new proof of its additive version. We further
introduce a third vertex of the $2$-category of regular toposes and show that
the diagram of $2$-(anti-)equivalences between three $2$-categories commutes,
the corresponding additive triangle is well-known.
