A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)

Descotte, María Emilia

doi:10.48550/arxiv.2010.10636

Cited by 1 publication

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“…This subsection deals with categorifications of the notion of filteredness: namely bifilteredness by [Ken92] and [Des20] and σ-filteredness by [DDS18], together with suited notion of 2-dimensional cofinality. Although those notions were developed separately, we show here they are actually almost the same in the sense that the seemingly more complicated σ-filtered σ-bicolimits automatically collapse on ordinary bifiltered bicolimits, which was seemingly unnoticed before.…”

Section: -Dimensional Filterednessmentioning

confidence: 99%

Bi-accessible and bipresentable 2-categories

Liberti¹,

Axel²

2022

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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 2dimensional colimits, filteredness and cofinality; in particular we show that σ-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 finitary 2-monads 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.

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Section: -Dimensional Filterednessmentioning

confidence: 99%