A note on coCartesian fibrations (2210.07753v1)

Abstract: We prove properness of (co)Cartesian fibrations as well as a straightening and unstraightening equivalence, which is compatible with cartesian products, when the base is the nerve of a small category.

• The paper proves that pullbacks along (co)Cartesian fibrations preserve Joyal equivalences using elementary combinatorial methods.
• The paper establishes a Quillen equivalence between marked simplicial set model structures for coCartesian and Cartesian fibrations under Joyal equivalences.
• The paper introduces directed univalence and universe properties that connect type theory with higher category theory in a novel way.

Analytical Overview of "A note on coCartesian fibrations" by Hoang Kim Nguyen

The paper "A note on coCartesian fibrations" authored by Hoang Kim Nguyen explores the field of higher category theory, focusing specifically on coCartesian fibrations. These structures are the higher categorical analogues of Grothendieck (op)fibrations, an essential tool in mathematical category theory. Nguyen addresses a straightening and unstraightening equivalence compatible with cartesian products when the base is the nerve of a small category. This establishes foundational work for a more comprehensive understanding of higher category structures and their practical applications.

Summary of Contributions

Nguyen's principal objective is to establish the properness of (co)Cartesian fibrations and to outline a streamlined proof of the straightening and unstraightening equivalence without relying on simplicial categories. This is necessary to validate essential properties of these fibrations, including compatibility with products and Joyal equivalences.

Key contributions of this work include:

1. Properness of Pullbacks: Nguyen proves that pullbacks along (co)Cartesian fibrations preserve Joyal equivalences. This extends existing literature, notably the works of Lurie, by providing proofs that do not depend on the full power of the straightening equivalence but rather use elementary methods.
2. Quillen Equivalence: The paper establishes that for any Joyal equivalence $A \to B$, there exists a Quillen equivalence for coCartesian and Cartesian model structures $sSet/A^\sharp \to sSet/B^\sharp$.
3. Directed Univalence and Universe Properties: In collaboration with Denis-Charles Cisinski, Nguyen introduces a new construction on the $\infty$-category of $\infty$-categories influenced by semantic models in type theory. The straightening equivalence follows from universe properties such as directed univalence, a novel complement to traditional methods.
4. Combinatorial Approach: Instead of using simplicial categories, the approach relies on combinatorial methods using marked simplicial sets to formulate the straightening/unstraightening theorem for nerves of categories. This provides a basis for expanding these concepts to broader $\infty$-categorical contexts.

Implications and Future Directions

Nguyen's work opens up several avenues for further exploration within both theoretical and applied mathematics:

• Broader Applications in Type Theory: By linking methods in type theory with higher category theory, this research paves the way for deeper synthesis in mathematical logic and computer science, particularly in proof theory and semantic models.
• Expansion to Simplicial Categories: While the paper avoids simplicial categories, further work could investigate a more expansive straightening/unstraightening theorem that includes them, potentially uncovering new results in $\infty$-category theory.
• Enhanced Proof Techniques: The methods and proofs outlined serve as templates for addressing other foundational problems in category theory using minimal assumptions or advanced equivalences.
• Potential for Algorithmic Implementations: Understanding these fibrations combinatorially and through nerve categories may lead to new computational models or algorithms, benefiting both software development and other data sciences needing categorical foundations.

In conclusion, Nguyen's paper is a robust contribution to higher category theory, crafting a refined understanding of coCartesian fibrations and establishing pathways for diverse research opportunities in mathematics and related fields. The treatment of their properties using straightforward, yet comprehensive methods sets a strong precedent for future studies in this intricate domain.