Combinatorics of higher-categorical diagrams

Abstract: This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent advances and practical experience with higher-dimensional diagram rewriting. We approach the subject as a kind of directed combinatorial topology: a diagram is a map from a "directed cell complex", encoded combinatorially as a face poset together with orientation data. Unlike previous expositions, we adopt from the beginning a functorial viewpoint, focussing on morphisms and categorical constructions. We do not tie ourselves to a specific model of higher categories, and instead treat diagrams as independent combinatorial structures that admit functorial interpretations in various contexts. Topics covered include the theory of layerings of diagrams; acyclicity properties and their consequences; constructions including Gray products, suspensions, and joins; special shapes such as globes, oriented simplices, cubes, and positive opetopes; the interpretation of diagrams in strict omega-categories and their geometric realisation as simplicial and CW complexes; and Steiner's theory of directed chain complexes.

• The paper introduces molecules as combinatorial representations of pasting diagrams, establishing unique isomorphisms and coherent composition laws.
• It employs pasting and rewrite construction methods to systematically build and analyze higher-categorical structures.
• The framework offers theoretical insights and practical advancements in category theory, algebraic topology, and automated reasoning tools.

An Overview of "Combinatorics of Higher-Categorical Diagrams"

The research paper "Combinatorics of Higher-Categorical Diagrams" by Amar Hadzihasanovic is an in-depth exploration of the mathematical structures known as higher-categorical diagrams. These diagrams play a pivotal role in modern mathematics, especially in areas like category theory, algebraic topology, and the foundations of quantum theory. The paper aims to systematically address the complexities and combinatorial properties of higher categories and diagrammatic reasoning.

Key Concepts and Definitions

At the heart of the paper is the concept of molecules, introduced as combinatorial structures representing shapes of pasting diagrams in higher-dimensional categories. Molecules are defined inductively, using two primary construction methods: pasting and rewrite construction. Both are operations on oriented graded posets—sets equipped with a grading and an orientation, resembling the structure of a higher-dimensional cell complex.

• Pasting: This operation combines two molecules along a shared substructure with a particular dimension (denoted as the k-boundary). The pasting process is analogous to the categorical operation of composition, ensuring that the composition of diagrams adheres to strict higher-category laws.
• Rewrite Construction: This construction method builds new molecules by combining two molecules, respecting their boundaries and treating the new cell as a higher-dimensional substitution.

Molecules are shown to be globular, meaning that their boundary operations interact coherently across dimensions, resembling globes in strict $\omega$-categories.

Results and Theorems

A significant portion of the paper is dedicated to proving that molecules possess unique isomorphisms, meaning they exhibit rigidity properties, akin to unique identities within strict higher categories. This leads to the conclusion that pasting satisfies associativity, unitality, and interchange laws up to unique isomorphism; properties essential for any coherent definition of a higher-dimensional category.

Furthermore, the concept of layerings is introduced—specific decompositions of molecules that simplify their complex structure into more manageable pieces. Layerings are employed for analyzing submolecules and identifying suitable substructures for rewriting operations. This is particularly crucial for practical applications like higher-dimensional rewriting systems and automated reasoning tools.

Implications and Future Developments

The implications of this research are twofold: theoretical and practical. Theoretically, it provides a robust combinatorial framework for understanding the nature of higher-categorical diagrams, advancing the fields of algebraic topology and category theory. Practically, these combinatorial insights lay groundwork for the implementation of efficient computational tools and proof assistants that handle higher-dimensional structures, offering potential advancements in both theoretical computer science and mathematical physics.

The paper suggests future research directions, particularly focusing on the expansion of these combinatorial methods to more general models of higher categories. One potential avenue is the exploration of weak higher categories, which could facilitate applications in homotopy theory and other domains where strict models become cumbersome.

Conclusion

Amar Hadzihasanovic's paper constructs a thorough and systematic treatment of the combinatorial properties of higher-categorical diagrams, offering valuable insight into the complexities underlying higher-dimensional diagrammatic reasoning. By establishing a foundational framework for understanding and manipulating such diagrams, the research opens doors to both theoretical advances and practical applications in computational mathematics and related fields.