A Categorical and Combinatorial Approach to Higher-Dimensional Diagrams and Their Applications

核心概念

This book introduces a novel framework for understanding higher-categorical diagrams, using the combinatorics of regular directed complexes and their morphisms to provide a powerful and flexible tool for reasoning about higher categories and their applications.

要約

Hadzihasanovic, A. (2024). Combinatorics of higher-categorical diagrams. [Book Manuscript].

This book aims to provide a comprehensive and modern treatment of higher-categorical diagrams, focusing on their combinatorial structure and functorial interpretations in various contexts.

抽出されたキーインサイト

Combinatorics of higher-categorical diagrams

by Amar Hadziha... 場所 arxiv.org 10-31-2024

https://arxiv.org/pdf/2404.07273.pdf

深掘り質問

How can the framework of regular directed complexes be extended to encompass diagrams in weak higher categories, where coherence conditions play a crucial role?

Extending the framework of regular directed complexes to weak higher categories presents a significant challenge, primarily due to the presence of coherence conditions. These conditions, often expressed as higher cells witnessing the equality of certain composites, are absent in the strict setting. Here's a breakdown of the challenges and potential approaches:
Challenges:

Representing Coherence: Regular directed complexes, as described in the context, rely on a strict notion of composition. Weak higher categories, however, necessitate a more nuanced approach. Coherence conditions, often represented as higher-dimensional cells, do not have a direct analogue in the current framework.
Composition and Coherence: The current definition of molecules as composable shapes needs to be re-evaluated. In weak higher categories, composition is not merely about 'gluing' shapes but also involves specifying coherence data.
Morphisms and Equivalences: The notions of maps and comaps might need refinement. In weak higher categories, we are often more interested in equivalences between diagrams rather than strict isomorphisms. This suggests exploring weaker notions of morphisms between regular directed complexes that incorporate coherence.
Potential Approaches:

Enriched Structures: One approach could be to enrich the combinatorial data of regular directed complexes. Instead of merely having sets of cells and boundary relations, we could consider enriching these sets over a category of "coherence witnesses." This could involve structures like simplicial sets or cubical sets, where higher-dimensional cells naturally encode coherence information.
Generalized Morphisms: Developing a more general notion of morphism between regular directed complexes is crucial. These morphisms should be able to accommodate the "rewriting" of diagrams up to coherence. This might involve incorporating ideas from homotopy theory or rewriting theory.
Axiomatic Approach: Instead of directly extending the existing definitions, an axiomatic approach could be fruitful. We could aim to identify key properties that a combinatorial framework for weak higher categories should satisfy. These properties might include closure under relevant constructions (like weak limits and colimits), a suitable notion of equivalence, and a way to interpret diagrams in different models of weak higher categories.
In summary, extending regular directed complexes to weak higher categories requires a careful rethinking of composition, coherence, and morphisms. While challenging, this endeavor holds the potential to provide a powerful tool for reasoning about and computing with weak higher structures.

Could alternative combinatorial structures, such as simplicial sets or cubical sets, provide a different perspective on higher-categorical diagrams and their applications?

Yes, alternative combinatorial structures like simplicial sets and cubical sets offer valuable perspectives on higher-categorical diagrams, complementing the framework of regular directed complexes. Here's a comparison:
Simplicial Sets:

Advantages: Simplicial sets excel at encoding compositionality and coherence. The simplicial structure naturally captures different ways of composing cells, and higher-dimensional simplices inherently represent coherence conditions. This makes them well-suited for modeling weak higher categories.
Perspective: Simplicial sets provide a more global perspective on diagrams. Instead of focusing on individual cells and their boundaries, they emphasize the relations and compositions between cells.
Applications: Simplicial sets are fundamental in homotopy theory and higher topos theory. They provide a powerful framework for studying homotopy coherent diagrams and constructing models of (∞,1)-categories.
Cubical Sets:

Advantages: Cubical sets are particularly well-suited for situations involving dualities and connections to geometry. The cubical structure naturally encodes dualities, and their geometric flavor makes them suitable for applications in concurrency theory and topological quantum computing.
Perspective: Cubical sets offer a more geometric and symmetric perspective on diagrams. The cubical structure lends itself well to visualizing higher-dimensional compositions.
Applications: Cubical sets are used in models of homotopy type theory, cubical higher category theory, and in the study of higher-dimensional automata.
Comparison with Regular Directed Complexes:

Focus: Regular directed complexes, as presented in the context, prioritize a local and strict perspective on diagrams, emphasizing the boundary structure of individual cells.
Compositionality: While regular directed complexes capture strict composition through pasting, simplicial and cubical sets provide a more flexible framework for handling compositions, including those governed by coherence conditions.
Applications: The choice of framework depends on the specific application. Regular directed complexes are well-suited for studying strict higher categories and higher-dimensional rewriting. Simplicial and cubical sets are more appropriate for exploring weak higher categories, homotopy coherence, and connections to other areas of mathematics.
In conclusion, simplicial and cubical sets offer valuable alternatives to regular directed complexes, providing complementary perspectives on higher-categorical diagrams. The choice of framework depends on the specific goals and the level of complexity required for a given application.

What are the implications of this combinatorial approach for the development of computational tools and proof assistants for higher category theory and related fields?

The combinatorial approach to higher-categorical diagrams, as exemplified by regular directed complexes, holds significant promise for developing computational tools and proof assistants in higher category theory and related fields. Here's an exploration of the implications:
Data Structures and Algorithms:

Concrete Representations: Combinatorial structures like regular directed complexes provide concrete, finitary data structures for representing higher-categorical diagrams. This makes them amenable to implementation in computers.
Algorithm Design: The well-defined nature of these structures enables the development of algorithms for manipulating and analyzing diagrams. For instance, algorithms for checking composability, performing pasting, identifying rewritable submolecules, and computing invariants of diagrams become feasible.
Efficiency: The combinatorial nature often leads to efficient algorithms. Many operations can be performed directly on the combinatorial data without resorting to complex topological or algebraic computations.
Proof Assistants and Formalization:

Formal Verification: The finite and combinatorial nature of these structures makes them suitable for formalization in proof assistants like Coq, Agda, or Lean. This allows for the formal verification of proofs and constructions involving higher-categorical diagrams.
Automated Reasoning: Proof assistants can be used to automate parts of the reasoning process. For example, checking the well-formedness of diagrams, verifying the applicability of rewriting rules, or even proving simple coherence conditions can be partially automated.
Increased Reliability: Formalization and automated reasoning contribute to increased reliability and trustworthiness of results in higher category theory, a field known for its intricate definitions and subtle arguments.
Applications and Impact:

Accessible Tools: Computational tools based on this approach can make higher category theory more accessible to a wider audience, including those without extensive background in the subject.
New Applications: The availability of computational tools can stimulate new applications of higher category theory in areas like computer science, physics, and other fields where diagrammatic reasoning is prevalent.
Bridging the Gap: This approach can help bridge the gap between theoretical developments in higher category theory and their practical applications, fostering a more symbiotic relationship between theory and practice.
Challenges and Future Directions:

Scalability: Developing tools that scale well to handle large and complex diagrams remains a challenge. Efficient algorithms and data structures are crucial for addressing this.
User Interfaces: Creating intuitive user interfaces for interacting with these tools is essential for wider adoption. Visualizations and graphical representations of diagrams can play a key role.
Integration with Existing Tools: Integrating these tools with existing proof assistants and mathematical software systems will enhance their utility and impact.
In conclusion, the combinatorial approach to higher-categorical diagrams has profound implications for developing computational tools and proof assistants. This promises to make the field more accessible, reliable, and applicable, potentially leading to breakthroughs in both theory and practice.