Bi-accessible and bipresentable 2-categories

We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that sigma-filteredness and bifilteredness are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of bicompact objects and bifiltered bicolimits. We then characterize them as categories of flat pseudofunctors. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of bifinitary pseudomonads on Cat are finitely bipresentable, which in particular captures the case of Lex, the 2-category of small lex categories. Invoking the technology of lex-colimits, we prove further that several 2-categories arising in categorical logic (Reg, Ex, Coh, Ext, Adh, Pretop) are also finitely bipresentable.