[Paper Review] An alternative characterisation of universal cells in opetopic n-categories

Eugenia Cheng|ArXiv.org|Apr 21, 2003

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper proposes an alternative characterization of universal cells in opetopic $n$-categories by defining composition with a $k$-cell as a span of $(n-k)$-categories, showing that a cell is universal if this span induces an equivalence. The key contribution is a new, more natural definition of universality that avoids arbitrary choices in composition, with equivalence to the original definition verified for $n \leq 2$. The approach aligns with the categorical principle that a morphism is an isomorphism iff composition with it is an isomorphism, adapted to the non-unique composition in opetopic $n$-categories.

ABSTRACT

We address the fact that composition in an opetopic weak n-category is in general not unique and hence is not a well-defined operation. We define composition with a given k-cell in an n-category by a span of (n-k)-categories. We characterise such a cell as universal if its composition span gives an equivalence of (n-k)-categories.

Motivation & Objective

• To address the non-uniqueness of composition in opetopic $n$-categories, which prevents defining composition as a well-defined operation.
• To provide an alternative, more natural characterization of universality that avoids arbitrary choices in composition.
• To generalize the categorical principle that a morphism is an isomorphism iff composition with it is an isomorphism to the setting of opetopic $n$-categories.
• To establish a new framework for universality using spans of $(n-k)$-categories, aligning with the spirit of weak higher categories.

Proposed method

• Define composition with a $k$-cell as a span of $(n-k)$-categories, capturing all possible ways of composing with the cell.
• Characterize a $k$-cell as universal if its composition span induces an equivalence of $(n-k)$-categories.
• Use the span-based definition to reframe the original notion of universality in opetopic $n$-categories.
• Verify the equivalence of the new and original definitions for $n \leq 2$ via explicit low-dimensional analysis.
• Employ a convention for ordering source elements in $k$-cells to represent symmetry classes, following Hermida, Makkai, and Power.
• Use pasting diagrams and factorization structures to model composition and universality in opetopic sets.

Experimental results

Research questions

• RQ1 Can universality in opetopic $n$-categories be redefined using spans of lower-dimensional categories instead of factorization conditions?
• RQ2 Does the span-based characterization of universality coincide with the original definition for low-dimensional cases ($n \leq 2$)?
• RQ3 Is the new characterization more natural or conceptually aligned with standard categorical principles, such as the isomorphism criterion via composition?
• RQ4 Can the span-based approach avoid the need for arbitrary choices in composition while preserving coherence in weak $n$-categories?
• RQ5 What is the relationship between the new universality condition and the original one in higher dimensions?

Key findings

• For $n=0$, the new and original definitions of universality coincide trivially, as all $k$-cells with $k > n$ are universal by definition.
• For $n=1$, the span-based characterization of universality is equivalent to the original definition: a 1-cell is universal iff the induced map on hom-sets is an isomorphism.
• For $n=2$, the new definition coincides with the original: a 1-cell is universal iff the induced map between categories of factorizations is an equivalence of 1-categories.
• The span-based approach successfully captures the essential structure of composition without requiring unique choices, aligning with the categorical principle that $f$ is an isomorphism iff $- \circ f$ is an isomorphism.
• The equivalence of the two definitions holds explicitly for $n \leq 2$, though a general proof for arbitrary dimensions remains open due to computational complexity.
• The new characterization is conceptually more satisfying and aligns better with standard categorical intuition, though it is less 'slick' in formal expression.