Información - ScientificComputing - # Higher Category Theory

An Equivalence Between Categorical Spectra and Pointed $(\infty, \mathbb{Z})$-Categories

Q: Could there be alternative categorical models for spectra that capture different aspects or generalizations of stable homotopy theory?

Yes, it's highly plausible that alternative categorical models for spectra exist, each potentially illuminating different facets or generalizations of stable homotopy theory. Here are some avenues to explore: Model Categories: While the paper focuses on an $\infty$-categorical approach, one could investigate model category structures on categories of diagrams or presheaves that give rise to equivalent homotopy categories of spectra. This could provide a more combinatorial or computational perspective. Higher Operads: Spectra are closely related to the linear structure encoded by the stable $\infty$-operad. Exploring models based on other higher operads or their generalizations could lead to "spectra" with different types of multiplicative structures, capturing different flavors of stable phenomena. Stable Derivators: Derivators provide an abstract framework for homotopy theory. It's conceivable that a suitable notion of a "stable derivator" could encapsulate the essential features of stable homotopy theory in a way that is independent of specific models like spectra. Type Theory: Homotopy type theory offers a foundational approach to homotopy theory. Exploring models for spectra within type theory could lead to new connections between stable homotopy theory and computer science, particularly in the realm of proof assistants and formal verification. Key Point: The search for alternative models is driven by the desire to understand stable homotopy theory from different angles and to extend its reach to new mathematical domains.

Q: What are the implications of this equivalence for the development of a more unified and conceptually satisfying framework for higher category theory and homotopy theory?

The equivalence between categorical spectra and pointed $(\infty, \mathbb{Z})$-categories has profound implications for the pursuit of a unified framework encompassing higher category theory and homotopy theory: Conceptual Simplification: The equivalence provides a more streamlined and conceptually satisfying understanding of categorical spectra, grounding them in the well-established language of $(\infty, \mathbb{Z})$-categories. This reduces the need for separate constructions and potentially simplifies proofs. Bridging the Gap: It strengthens the bridge between the traditionally distinct worlds of higher category theory and stable homotopy theory. This can foster the transfer of ideas and techniques between these areas, leading to new insights and progress on both sides. New Foundations: The algebraic nature of $(\infty, \mathbb{Z})$-categories might offer a more foundational perspective on stable homotopy theory. This could lead to new axiomatic frameworks for stable homotopy theory that are more closely aligned with the principles of higher category theory. Broader Applications: A unified framework could make the powerful tools of stable homotopy theory more accessible to mathematicians working in other fields. This could lead to new applications and connections that were previously unforeseen. Key Point: This equivalence is a significant step towards a more integrated and conceptually elegant understanding of higher structures in mathematics, with the potential to unlock new discoveries and connections across various fields.

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Categorical spectra, a higher-categorical analogue of spectra, are equivalent to pointed $(\infty, \mathbb{Z})$-categories, providing a more algebraic understanding of these stable structures.

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arxiv.org

Kern, D. (2024, October 4). Categorical spectra as pointed (∞, Z)-categories [Preprint]. arXiv:2410.02578v1 [math.CT]

This paper aims to establish a concrete connection between categorical spectra and  $(\infty, \mathbb{Z})$-categories, clarifying the algebraic nature of these stable homotopy-theoretic objects.

Ideas clave extraídas de

Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories

by David Kern a las arxiv.org 10-04-2024

https://arxiv.org/pdf/2410.02578.pdf

$Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories$

Consultas más profundas

How might this new understanding of categorical spectra as pointed $(\infty, \mathbb{Z})$-categories be applied to other areas of mathematics where stable homotopy theory plays a role, such as algebraic geometry or algebraic topology?

This new perspective on categorical spectra as pointed $(\infty, \mathbb{Z})$-categories offers exciting potential applications in areas where stable homotopy theory intersects with other mathematical disciplines. Here are some possibilities:
Algebraic Geometry:

Motivic Homotopy Theory:  Motivic homotopy theory replaces the sphere spectrum with a "motivic sphere spectrum" that encodes arithmetic and geometric information about a scheme. The equivalence between categorical spectra and pointed $(\infty, \mathbb{Z})$-categories could lead to new models or insights into motivic spectra, potentially revealing deeper connections between algebraic geometry and higher category theory.
Derived Algebraic Geometry:  In derived algebraic geometry, one works with "derived" versions of geometric objects, which carry richer homotopical information. The algebraic nature of $(\infty, \mathbb{Z})$-categories might provide a convenient framework for studying derived categories of sheaves and other derived structures arising in this context.
Algebraic Topology:

Equivariant Stable Homotopy Theory:  Equivariant stable homotopy theory studies spaces with group actions. The structure of $(\infty, \mathbb{Z})$-categories, particularly the interplay between objects and morphisms across different dimensions, could be leveraged to develop models for equivariant spectra that are more directly amenable to categorical techniques.
Generalized Cohomology Theories:  Categorical spectra provide a natural setting for understanding generalized cohomology theories. The equivalence with pointed $(\infty, \mathbb{Z})$-categories might offer new tools for constructing and analyzing such theories, potentially leading to new invariants and computational methods.
Key Point: The algebraic nature of $(\infty, \mathbb{Z})$-categories makes them well-suited for studying structures with rich algebraic and homotopical data. This opens up avenues for applying stable homotopy theory in contexts where such data naturally arises.

Could there be alternative categorical models for spectra that capture different aspects or generalizations of stable homotopy theory?

Yes, it's highly plausible that alternative categorical models for spectra exist, each potentially illuminating different facets or generalizations of stable homotopy theory. Here are some avenues to explore:

Model Categories: While the paper focuses on an $\infty$-categorical approach, one could investigate model category structures on categories of diagrams or presheaves that give rise to equivalent homotopy categories of spectra. This could provide a more combinatorial or computational perspective.
Higher Operads:  Spectra are closely related to the linear structure encoded by the stable $\infty$-operad. Exploring models based on other higher operads or their generalizations could lead to "spectra" with different types of multiplicative structures, capturing different flavors of stable phenomena.
Stable Derivators: Derivators provide an abstract framework for homotopy theory. It's conceivable that a suitable notion of a "stable derivator" could encapsulate the essential features of stable homotopy theory in a way that is independent of specific models like spectra.
Type Theory: Homotopy type theory offers a foundational approach to homotopy theory. Exploring models for spectra within type theory could lead to new connections between stable homotopy theory and computer science, particularly in the realm of proof assistants and formal verification.
Key Point: The search for alternative models is driven by the desire to understand stable homotopy theory from different angles and to extend its reach to new mathematical domains.

What are the implications of this equivalence for the development of a more unified and conceptually satisfying framework for higher category theory and homotopy theory?

The equivalence between categorical spectra and pointed $(\infty, \mathbb{Z})$-categories has profound implications for the pursuit of a unified framework encompassing higher category theory and homotopy theory:

Conceptual Simplification: The equivalence provides a more streamlined and conceptually satisfying understanding of categorical spectra, grounding them in the well-established language of $(\infty, \mathbb{Z})$-categories. This reduces the need for separate constructions and potentially simplifies proofs.
Bridging the Gap: It strengthens the bridge between the traditionally distinct worlds of higher category theory and stable homotopy theory. This can foster the transfer of ideas and techniques between these areas, leading to new insights and progress on both sides.
New Foundations: The algebraic nature of $(\infty, \mathbb{Z})$-categories might offer a more foundational perspective on stable homotopy theory. This could lead to new axiomatic frameworks for stable homotopy theory that are more closely aligned with the principles of higher category theory.
Broader Applications: A unified framework could make the powerful tools of stable homotopy theory more accessible to mathematicians working in other fields. This could lead to new applications and connections that were previously unforeseen.
Key Point: This equivalence is a significant step towards a more integrated and conceptually elegant understanding of higher structures in mathematics, with the potential to unlock new discoveries and connections across various fields.