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First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\sigma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c3<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-<jats:italic>filteredness<\/jats:italic> and <jats:italic>bifilteredness<\/jats:italic> are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of <jats:italic>bicompact<\/jats:italic> objects and <jats:italic>bifiltered<\/jats:italic> bicolimits. We then characterize them as categories of <jats:italic>flat pseudofunctors<\/jats:italic>. We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small <jats:italic>bilex<\/jats:italic> 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of finitary 2-monads on <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{Cat}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>Cat<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> are finitely bipresentable, which in particular captures the case of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{Lex}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>Lex<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, the 2-category of small lex categories. Invoking the technology of <jats:italic>lex-colimits<\/jats:italic>, we prove further that several 2-categories arising in categorical logic (<jats:bold>Reg, Ex, Coh, Ext, Adh, Pretop<\/jats:bold>) are also finitely bipresentable.\n<\/jats:p>","DOI":"10.1007\/s10485-024-09794-9","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T09:07:51Z","timestamp":1733735271000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bi-accessible and Bipresentable 2-Categories"],"prefix":"10.1007","volume":"33","author":[{"given":"Ivan","family":"Di Liberti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Axel","family":"Osmond","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"9794_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600579","volume-title":"Locally Presentable and Accessible Categories","author":"J Adamek","year":"1994","unstructured":"Adamek, J., Rosicky, J.: Locally Presentable and Accessible Categories, vol. 189. 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