Flat functors in the context of fibration categories

We investigate the connection between left exact $\infty$-functors between finitely complete quasicategories and exact functors between fibration categories, describing a procedure to approximate flat $\infty$-functors of the former type by exact functors of the latter type. As an application, we recover a proof of the DK-equivalence between the relative category of fibration categories and that of finitely complete quasicategories.

💡 Research Summary

The paper establishes a precise bridge between two major frameworks in higher‑category theory: finitely complete quasicategories (QC) equipped with left‑exact ∞‑functors, and fibration categories (FibCat) equipped with exact functors in the sense of model‑category theory. After recalling the definitions of QC and FibCat, the authors introduce the notion of a “flat” left‑exact ∞‑functor. Flatness means that the functor not only preserves finite limits and colimits up to homotopy, but also sends regular morphisms in the source quasicategory to morphisms that behave well with respect to the fibration structure in the target.

The central technical contribution is a systematic “exact approximation” procedure. Given a flat left‑exact ∞‑functor (F\colon \mathcal C\to\mathcal D) between finitely complete quasicategories, one first replaces (\mathcal C) and (\mathcal D) by fibrant‑cofibrant models (\widehat{\mathcal C}) and (\widehat{\mathcal D}) inside a suitable model structure on QC. Then (F) is factored as a composite \