(∞, 2)-Topoi and Descent: A Characterization via Fibrational Descent and its Implications for Synthetic (∞, 1)-Category Theory

Q: How does the theory of (∞, 2)-topoi inform and potentially enrich our understanding of homotopy type theory and its models?

Answer: The theory of (∞, 2)-topoi offers a powerful new perspective on homotopy type theory and its models by providing a framework where directedness is intrinsic. Here's how: Synthetic (∞, 1)-Categories: While homotopy type theory focuses on synthetic ∞-groupoids, (∞, 2)-topoi allow us to work directly with synthetic (∞, 1)-categories. This means we can reason about objects with non-invertible morphisms internally, capturing the essence of directed homotopy theory. Fibrational Descent and the Classifying Object: The central axiom of fibrational descent in an (∞, 2)-topos ensures the existence of a classifying object for internal fibrations. This is analogous to the universe object in homotopy type theory, which classifies types. This suggests a deep connection between the two settings, potentially leading to new models of homotopy type theory within (∞, 2)-topoi. Directed Univalence: The directed univalence principle in (∞, 2)-topoi mirrors the univalence axiom of homotopy type theory but in a directed setting. It establishes a coherence between the internal mapping groupoid of classifying objects and the groupoid core of the synthetic functor category. This hints at a richer, more nuanced notion of equivalence in the directed setting. Limitations and New Insights: Importantly, not all of homotopy type theory translates directly into the (∞, 2)-topos context. The existence of objects with empty groupoid cores, for example, reveals limitations in applying object-wise arguments. This highlights the unique features of directed homotopy theory and prompts a re-examination of familiar concepts in a new light. In summary, (∞, 2)-topoi provide a bridge between traditional homotopy type theory and directed homotopy theory. This connection can lead to new models, a deeper understanding of directed homotopy types, and a richer perspective on the foundations of homotopy-coherent mathematics.

Q: Could the concept of "fibrational descent" be further generalized to define and study even higher-categorical analogs of topoi, such as (∞, n)-topoi for n > 2?

Answer: Yes, the concept of "fibrational descent" appears ripe for generalization to higher categorical settings, potentially leading to a theory of (∞, n)-topoi. Here's a plausible roadmap: Higher Fibrations: The first step would involve developing a robust theory of higher fibrations within (∞, n)-categories. This would likely build upon existing notions of (∞, n)-categorical fibrations and require careful consideration of the various possible flavors of (op)laxness in higher categories. Generalized Oriented Pullbacks: The notion of oriented pullbacks, crucial for defining internal fibrations in (∞, 2)-topoi, would need to be generalized. This might involve identifying specific (op)lax limits in (∞, n)-categories that capture the desired properties of higher fibrations. (∞, n)-Fibrational Descent: With these ingredients in place, one could formulate an (∞, n)-categorical version of fibrational descent. This axiom would likely assert that certain functors, analogous to Fib0 and Fib1, preserve appropriate classes of (op)lax limits. Challenges and Potential: This generalization is not without its challenges. The increasing complexity of higher categories demands careful consideration of coherence issues and the interplay between different levels of (op)laxness. However, the potential rewards are significant. A well-developed theory of (∞, n)-topoi could provide: New Foundations: A foundation for synthetic (∞, (n-1))-category theory. Higher Homotopy Theory: Deeper insights into higher homotopy theory and its relationship to higher category theory. Generalized Logic: Connections to higher-categorical logic and type theory. In conclusion, generalizing fibrational descent to (∞, n)-topoi is a promising avenue for future research. While challenging, it holds the potential to significantly advance our understanding of higher categories and their applications.

Q: What are the potential implications of this framework for developing a deeper understanding of the relationship between directed homotopy theory and traditional homotopy theory?

Answer: The framework of (∞, 2)-topoi offers a fertile ground for exploring the interplay between directed and traditional homotopy theory. Here are some potential implications: Unified Perspective: (∞, 2)-Topoi provide a common language for both directed and traditional homotopy theory. By viewing ∞-groupoids as special cases of (∞, 1)-categories, we can study both within the same overarching framework. Directed Approximations: The (∞, 1)-localic approximation of an (∞, 2)-topos, as described by the adjunction with ShCat(∞,1)(−), provides a way to approximate directed homotopy types using traditional homotopy types. This could lead to new tools for studying directed homotopy theory using techniques from the well-established realm of ∞-groupoids. New Invariants: The study of (∞, 2)-topoi might lead to new invariants of directed homotopy types, going beyond the usual invariants of ∞-groupoids. These invariants could capture subtle aspects of directedness and provide a finer classification of such spaces. Coherence and Directedness: The challenges posed by objects with empty groupoid cores highlight the importance of coherence in directed homotopy theory. Understanding these obstructions could lead to a deeper appreciation of the role of coherence in higher categories and its relationship to directedness. Applications Beyond Topology: The interplay between directed and traditional homotopy theory within (∞, 2)-topoi could have implications for other areas of mathematics where directed structures arise, such as concurrency theory, higher category theory, and even theoretical physics. In conclusion, the framework of (∞, 2)-topoi offers a powerful lens through which to investigate the relationship between directed and traditional homotopy theory. This line of inquiry has the potential to uncover new connections, deepen our understanding of both subjects, and lead to applications in various areas of mathematics.

Grunnleggende konsepter

This paper introduces the concept of (∞, 2)-topoi, a higher-categorical generalization of topoi, characterized by a "fibrational descent" axiom, and explores its implications for developing a synthetic theory of (∞, 1)-categories.

Sammendrag

Bibliographic Information

Abellán, F., & Martini, L. (2024, October 2). (∞, 2)-Topoi and descent. arXiv. https://arxiv.org/abs/2410.02014v1

Research Objective

This paper aims to establish a foundational framework for (∞, 2)-topoi, higher-categorical analogs of topoi, by introducing the concept of "fibrational descent" as a defining axiom. The authors investigate the implications of this axiom for developing a synthetic theory of (∞, 1)-categories within this framework.

Methodology

The authors employ methods from higher category theory, particularly (∞, 2)-category theory, to define and explore the properties of (∞, 2)-topoi. They introduce the notion of "fibrational descent," inspired by the descent axiom for (∞, 1)-topoi, and use it to characterize (∞, 2)-topoi. They further develop the theory of internal fibrations, partially lax Kan extensions, and the Yoneda embedding within this context.

Key Findings

The authors introduce the axiom of "fibrational descent" for presentable (∞, 2)-categories, which asserts that certain functors classifying internal fibrations preserve specific types of limits.
They prove a 2-dimensional version of Giraud's theorem, characterizing (∞, 2)-topoi as localizations of presheaf categories preserving "oriented pullbacks."
The paper establishes that (∞, 2)-topoi admit an internal theory of (∞, 1)-categories, including notions of ∞-groupoids, directed univalence, synthetic Kan extensions, and a Yoneda embedding.
The authors demonstrate that the theory of (∞, 1)-topoi embeds as a full sub-2-category of the (∞, 2)-category of (∞, 2)-topoi, with the embedding given by considering sheaves of (∞, 1)-categories.

Main Conclusions

The paper successfully establishes a robust framework for (∞, 2)-topoi based on the fibrational descent axiom. This framework provides a natural setting for developing a synthetic theory of (∞, 1)-categories, offering new insights into their behavior and interactions. The results suggest that (∞, 2)-topoi provide a rich and nuanced perspective on higher category theory and its applications.

Significance

This work significantly contributes to the field of higher category theory by providing a solid foundation for (∞, 2)-topoi and their connection to synthetic (∞, 1)-category theory. It opens up new avenues for research in areas such as homotopy type theory, higher topos theory, and their applications to other mathematical disciplines.

Limitations and Future Research

The paper primarily focuses on establishing the foundational aspects of (∞, 2)-topoi. Further research could explore more advanced topics within this framework, such as the development of cohomology theories, the study of geometric structures, and the exploration of potential applications in areas like algebraic geometry and mathematical physics.

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$(\infty,2)$-Topoi and descent

by Fern... klokken arxiv.org 10-04-2024

https://arxiv.org/pdf/2410.02014.pdf

$$(\infty,2)$-Topoi and descent$

Dypere Spørsmål

How does the theory of (∞, 2)-topoi inform and potentially enrich our understanding of homotopy type theory and its models?

Answer:
The theory of (∞, 2)-topoi offers a powerful new perspective on homotopy type theory and its models by providing a framework where directedness is intrinsic. Here's how:

Synthetic (∞, 1)-Categories:  While homotopy type theory focuses on synthetic ∞-groupoids, (∞, 2)-topoi allow us to work directly with synthetic (∞, 1)-categories. This means we can reason about objects with non-invertible morphisms internally, capturing the essence of directed homotopy theory.

Fibrational Descent and the Classifying Object: The central axiom of fibrational descent in an (∞, 2)-topos ensures the existence of a classifying object for internal fibrations. This is analogous to the universe object in homotopy type theory, which classifies types. This suggests a deep connection between the two settings, potentially leading to new models of homotopy type theory within (∞, 2)-topoi.

Directed Univalence:  The directed univalence principle in (∞, 2)-topoi mirrors the univalence axiom of homotopy type theory but in a directed setting. It establishes a coherence between the internal mapping groupoid of classifying objects and the groupoid core of the synthetic functor category. This hints at a richer, more nuanced notion of equivalence in the directed setting.

Limitations and New Insights: Importantly, not all of homotopy type theory translates directly into the (∞, 2)-topos context. The existence of objects with empty groupoid cores, for example, reveals limitations in applying object-wise arguments. This highlights the unique features of directed homotopy theory and prompts a re-examination of familiar concepts in a new light.
In summary, (∞, 2)-topoi provide a bridge between traditional homotopy type theory and directed homotopy theory. This connection can lead to new models, a deeper understanding of directed homotopy types, and a richer perspective on the foundations of homotopy-coherent mathematics.

Could the concept of "fibrational descent" be further generalized to define and study even higher-categorical analogs of topoi, such as (∞, n)-topoi for n > 2?

Answer:
Yes, the concept of "fibrational descent" appears ripe for generalization to higher categorical settings, potentially leading to a theory of (∞, n)-topoi. Here's a plausible roadmap:

Higher Fibrations: The first step would involve developing a robust theory of higher fibrations within (∞, n)-categories. This would likely build upon existing notions of (∞, n)-categorical fibrations and require careful consideration of the various possible flavors of (op)laxness in higher categories.

Generalized Oriented Pullbacks:  The notion of oriented pullbacks, crucial for defining internal fibrations in (∞, 2)-topoi, would need to be generalized. This might involve identifying specific (op)lax limits in (∞, n)-categories that capture the desired properties of higher fibrations.

(∞, n)-Fibrational Descent: With these ingredients in place, one could formulate an (∞, n)-categorical version of fibrational descent. This axiom would likely assert that certain functors, analogous to Fib0 and Fib1, preserve appropriate classes of (op)lax limits.

Challenges and Potential:  This generalization is not without its challenges. The increasing complexity of higher categories demands careful consideration of coherence issues and the interplay between different levels of (op)laxness. However, the potential rewards are significant. A well-developed theory of (∞, n)-topoi could provide:

New Foundations: A foundation for synthetic (∞, (n-1))-category theory.
Higher Homotopy Theory:  Deeper insights into higher homotopy theory and its relationship to higher category theory.
Generalized Logic:  Connections to higher-categorical logic and type theory.
In conclusion, generalizing fibrational descent to (∞, n)-topoi is a promising avenue for future research. While challenging, it holds the potential to significantly advance our understanding of higher categories and their applications.

What are the potential implications of this framework for developing a deeper understanding of the relationship between directed homotopy theory and traditional homotopy theory?

Answer:
The framework of (∞, 2)-topoi offers a fertile ground for exploring the interplay between directed and traditional homotopy theory. Here are some potential implications:

Unified Perspective: (∞, 2)-Topoi provide a common language for both directed and traditional homotopy theory. By viewing ∞-groupoids as special cases of (∞, 1)-categories, we can study both within the same overarching framework.

Directed Approximations: The (∞, 1)-localic approximation of an (∞, 2)-topos, as described by the adjunction with ShCat(∞,1)(−), provides a way to approximate directed homotopy types using traditional homotopy types. This could lead to new tools for studying directed homotopy theory using techniques from the well-established realm of ∞-groupoids.

New Invariants: The study of (∞, 2)-topoi might lead to new invariants of directed homotopy types, going beyond the usual invariants of ∞-groupoids. These invariants could capture subtle aspects of directedness and provide a finer classification of such spaces.

Coherence and Directedness: The challenges posed by objects with empty groupoid cores highlight the importance of coherence in directed homotopy theory. Understanding these obstructions could lead to a deeper appreciation of the role of coherence in higher categories and its relationship to directedness.

Applications Beyond Topology: The interplay between directed and traditional homotopy theory within (∞, 2)-topoi could have implications for other areas of mathematics where directed structures arise, such as concurrency theory, higher category theory, and even theoretical physics.
In conclusion, the framework of (∞, 2)-topoi offers a powerful lens through which to investigate the relationship between directed and traditional homotopy theory. This line of inquiry has the potential to uncover new connections, deepen our understanding of both subjects, and lead to applications in various areas of mathematics.