Bi-accessible and bipresentable 2-categories
Ivan Di Liberti, Axel Osmond

TL;DR
This paper extends the concepts of accessibility and presentability to 2-categories, establishing a framework for bi-accessibility and bipresentability, characterizing them via flat pseudofunctors, and proving duality and representability results.
Contribution
It introduces bi-accessible and bipresentable 2-categories, characterizes them through flat pseudofunctors, and establishes a 2-dimensional Gabriel-Ulmer duality.
Findings
Bi-accessible and bipresentable 2-categories are characterized via flat pseudofunctors.
A bi-accessible right bi-adjoint functor theorem is proved.
Several 2-categories in categorical logic are shown to be finitely bipresentable.
Abstract
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…
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Taxonomy
TopicsAdvanced Algebra and Logic · Fuzzy and Soft Set Theory · Rings, Modules, and Algebras
