Overview
This work explores the theoretical framework of 2-categories and elementary 2-topoi, building upon Weber's established notion. The central objective involves the introduction and application of specific new axioms to elaborate on the structure and properties of these mathematical constructs. A primary finding concerns the behavior of the "discrete opfibration (DOF) classifier" (S) within a 2-category that adheres to these proposed axioms.
Furthermore, the research investigates the role of "groupoidal" objects within this advanced categorical context. It posits that these objects are fundamental to the overall structure of certain 2-categories and presents formal results supporting this claim.
Research Context
The research originates from the existing concept of an elementary 2-topos, as initially introduced by Weber. This foundational concept serves as the starting point for the current investigation, which aims to extend and refine the understanding of these structures through a series of new axiomatic conditions. The development of these axioms is directed toward uncovering deeper relationships and properties within 2-categories, particularly concerning the interaction between discrete opfibrations and 1-topoi.
Approach
The approach involves the proposal of specific new axioms intended to govern the behavior of 2-categories. These axioms are characterized as closure conditions that apply to discrete opfibrations (DOFs) having "S-small fibres." A notable closure condition among these new axioms relates to a DOF that is described as analogous to the "subset fibration" over Set, stipulating that this particular DOF must have small fibres.
Additional axioms introduced focus on "groupoidal" objects within a 2-category. The research utilizes these axioms to establish connections between the global structure of a 2-category and its groupoidal components. The methodology involves demonstrating how these axioms lead to structural determinations within the 2-category.
Findings
The research establishes several key findings:
- In a 2-category (C) that satisfies the newly proposed axioms, the "discrete opfibration (DOF) classifier" (S) is consistently an internal elementary 1-topos. This holds true in an appropriate sense defined by the framework.
- The axioms introduced are described as closure conditions on DOFs that possess "S-small fibres."
- One specific closure condition, highlighted for its interest, asserts that a certain DOF – which is analogous to the "subset fibration" over Set – must have small fibres.
- New axioms were also introduced concerning "groupoidal" objects in a 2-category, indicating their significant role in the general theory.
- The research presents two distinct results illustrating that a 2-category (C) satisfying these axioms is "determined by" its groupoidal objects.
- The first of these results demonstrates that (C) is equivalent to a 2-category of internal categories that are constructed out of groupoidal objects.
- The second result indicates that the groupoidal objects are dense within (C).
Why This Matters
The development of new axioms for elementary 2-topoi enhances the theoretical understanding of higher-dimensional categorical structures. Identifying the discrete opfibration classifier as an internal elementary 1-topos provides a specific structural insight within these complex mathematical systems. The demonstration that 2-categories can be determined by their groupoidal objects highlights the foundational importance of these particular objects in the broader theory of 2-categories, potentially simplifying the study of these structures by focusing on key components.
Key Limitations Mentioned by Researchers
The source document does not explicitly mention any limitations of the research.