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Flat functors in the context of fibration categories

El Mehdi Cherradi

TL;DR

The paper builds a bridge between left exact $\infty$-functors and exact functors between fibration categories by rigidifying flat $\infty$-functors into spans of exact functors that model left Kan extensions. It develops a Yoneda-style embedding and a suite of technical tools to translate homotopical data into strict presentations, enabling a DK-equivalence proof between the relative category of fibration categories and finitely complete quasicategories. The approach is then extended to tribes, showing the broader applicability of the rigidification technique to multiple models of higher categories. Overall, the work provides a concrete, functorial pathway from $(\infty,1)$-categorical flatness to 1-categorical exactness, with explicit constructions and implications for equivalences of homotopy theories.

Abstract

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.

Flat functors in the context of fibration categories

TL;DR

The paper builds a bridge between left exact -functors and exact functors between fibration categories by rigidifying flat -functors into spans of exact functors that model left Kan extensions. It develops a Yoneda-style embedding and a suite of technical tools to translate homotopical data into strict presentations, enabling a DK-equivalence proof between the relative category of fibration categories and finitely complete quasicategories. The approach is then extended to tribes, showing the broader applicability of the rigidification technique to multiple models of higher categories. Overall, the work provides a concrete, functorial pathway from -categorical flatness to 1-categorical exactness, with explicit constructions and implications for equivalences of homotopy theories.

Abstract

We investigate the connection between left exact -functors between finitely complete quasicategories and exact functors between fibration categories, describing a procedure to approximate flat -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.
Paper Structure (16 sections, 25 theorems, 58 equations)

This paper contains 16 sections, 25 theorems, 58 equations.

Table of Contents

  1. Fibration categories and exact functors
  2. The Yoneda embedding
  3. Some technical tools
  4. From filtered quasicategories to filtered categories
  5. Fibration categories in $\mathbf{RelCat}$
  6. Projective cofibrancy from Reedy cofibrancy
  7. Rigidifying flat $\infty$-functors
  8. The unparameterized case
  9. The general case
  10. Rigid left Kan extension
  11. The unparameterized case
  12. The general case
  13. Applications
  14. Extension to tribes
  15. The Yoneda embedding
  16. ...and 1 more sections

Key Result

Lemma 1.1

Let $F : \mathcal{C} \to \mathcal{D}$ be a functor between fibration categories. Assume that $F$ maps pullbacks along fibrations to pullback squares that are homotopy pullbacks, that $F(*_\mathcal{C}) \to *_\mathcal{D}$ is a weak equivalence and that weak equivalences are sent to weak equivalences b

Theorems & Definitions (64)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.1
  • Lemma 1.1
  • proof
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Proposition 2.1
  • ...and 54 more