Structural Results on Bifibrations of (∞, 1)-Categories with Internal Sums

Core Concepts

The author presents structural results on bifibrations of (∞, 1)-categories with internal sums, characterizing cartesian bifibrations with extensive internal sums as Artin gluings of lex functors. The approach follows Streicher's presentation of fibered category theory in a synthetic setting.

Abstract

The content delves into the characterization of internal sums for fibrations of synthetic (∞, 1)-categories. It explores Moens' Theorem and its higher version, along with generalized versions due to Streicher. The study extends to the notion of extensivity and its implications in higher topos theory from a fibrational perspective. Various theorems and propositions are discussed regarding cocartesian families, Beck–Chevalley conditions, and Moens fibrations.
Key points include:

Characterization of extensive internal sums in categories.
Application to geometric morphisms between lex categories.
Generalizations and equivalences related to Moens' Theorem.
Closure properties and naturality results for cocartesian families and functors.
The content provides insights into advanced mathematical concepts related to fibered categories and their applications in higher categorical logic.

Stats

Classically, a category C with all small coproducts is called (infinitary) extensive if for any small family of objects (ai)i∈I in C the canonical maps Q i∈I C/ai → C/ ∑ i∈I ai.
A co-/cartesian family P : B → U of U-small synthetic (∞, 1)-categories possesses functorial transport for directed arrows in the base.
A Beck–Chevalley family is a bicartesian family that satisfies the Beck–Chevalley condition (BCC), denoting that any dependent square over a pullback is itself a pullback precisely if the top horizontal map is cocartesian.

Quotes

"Moens’ Theorem establishes a correspondence between lex functors and lextensive fibration." - Jonathan Weinberger

Key Insights Distilled From

Internal sums for synthetic fibered $(\infty,1)$-categories

by Jonathan Wei... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2205.00386.pdf

Deeper Inquiries

How does the concept of extensivity impact the study of higher topos theory?

Extensivity plays a crucial role in the study of higher topos theory as it provides a key characterization for certain types of fibrations. In particular, extensivity is closely related to the descent property, which is fundamental in Giraud's characterization of Grothendieck toposes. Extensive fibrations are essential for understanding geometric morphisms between lex categories and have implications for internal higher topos theory.
By studying extensivity in fibered (∞, 1)-categories, researchers can gain insights into how categorical structures behave with respect to pullbacks and coproducts. This allows for a deeper understanding of how different categories interact within an (∞, 1)-topos setting. The concept of extensivity helps establish relationships between various types of fibrations and provides a framework for analyzing geometric properties within higher category theory.
In summary, extensivity serves as a foundational concept that underpins many aspects of higher topos theory by providing a way to characterize important properties of fibrations and their relationship with other categorical structures.

What are potential applications of Moens' Theorem beyond geometric morphisms?

Moens' Theorem has broader applications beyond just geometric morphisms in fibered category theory. Some potential applications include:

Higher Categorical Logic: Moens' Theorem can be used as a tool in higher categorical logic studies where characterizing fibrations with extensive internal sums is essential for reasoning about logical systems involving internal (∞, 1)-categories.

Model-Independent Theory: The theorem could potentially be extended to model-independent versions applicable across different settings such as arbitrary ∞-cosmoses or even (∞, n)-categories for varying values of n.

Topos Theory Generalizations: By extending Moen's results into more generalized contexts like (∞, 1)-toposes or elementary cases à la Shulman–Rasekh frameworks, new insights into topological spaces at different levels could be gained.

Categorical Modal Logic: Applications in modal logic using fibration viewpoints similar to those explored by Doat could benefit from leveraging Moen's results in developing logical systems based on categorial principles.

How does the closure property under composition affect the analysis...

The closure property under composition significantly impacts the analysis and understanding...
[Add detailed response explaining how this closure property influences cocartesian families]