On the Unicity of the Homotopy Theory of Higher Categories (1112.0040v6)

Abstract: We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces, Simpson and Tamsamani's Segal $n$-categories, the first author's $n$-fold complete Segal spaces, Kan and the first author's $n$-relative categories, and complete Segal space objects in any model of $(\infty,n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb{Z}/2)^n$.

• The paper introduces four axioms that uniquely characterize the homotopy theory of (∞,n)-categories, ensuring consistency across various models.
• It demonstrates that the space of such quasicategories is homotopy equivalent to (RP∞)^n, unifying previously disparate approaches.
• The framework reconciles models like Rezk’s complete Segal spaces and Lurie’s theories, confirming and expanding upon earlier conjectures in higher category theory.

On the Unicity of the Homotopy Theory of Higher Categories

The paper "On the Unicity of the Homotopy Theory of Higher Categories" by Clark Barwick and Christopher Schommer-Pries embarks on establishing a coherent and comprehensive framework for the homotopy theory of $(\infty,n)$-categories. Higher category theory has ventured through a tumultuous developmental history, offering a plethora of models aimed at capturing the underlying structure of operations and transformations at different levels of abstraction. This paper is distinct in its efforts to form an axiomatic foundation for these categories, drawing from a universal property that informs and encompasses the myriad definitions proposed over time.

The authors introduce four axioms that any reasonable homotopy theory of $(\infty,n)$-categories must satisfy. A compelling aspect of their work is the proof that the patchwork of many theories and models can be cohesively brought under a single framework, showing that the space of such quasicategories is homotopy equivalent to $(RP^\infty)^n$. This remarkable result implies that all such categories, if defined correctly, are "essentially" the same up to certain specific transformations—highlighting the unicity of homotopy theories under the structure provided.

Key contributions of the paper include:

1. Axioms for Homotopy Theories: The authors propose axioms for homotopy theories of $(\infty,n)$-categories—Strong Generation, Internal Homs for Correspondences, Fundamental Pushouts, and Universality—ensuring that the homotopy theory uniquely characterizes $(\infty,n)$-categories.
2. Generations and Equivalences: Strong Generation insists that the category $C$ is generated from a chosen subcategory closed under homotopy colimits. Internal Homs for Correspondences ensures that the theory extends naturally to categories of correspondences, providing a deeper structural insight into the nature of higher morphisms.
3. Fundamental Pushouts: This axiom imposes the necessity of a finite list of pushouts which do not distort the intended homotopy theory—the foundation to ensure $C$ exhibits desirable categorical properties.
4. Implications and Examples: Notably, the paper shows the equivalence of well-known models such as Rezk’s complete Segal spaces and $n$-fold complete Segal spaces through specific recognition theorems. Several model categories, including the examples by Joyal, Kan, and Lurie, comply with the proposed axioms.

The authors also subtractively leverage examples to confirm the viability of their axiomatic approach. By proving that existing models fit seamlessly into this refined framework, the work notably synthesizes and verifies prior conjectures, notably those by Simpson, and generalizations of Toën’s results.

The paper presents a sophisticated interplay between category theory, topology, and homotopy, addressing practical implications and unifying disparate theoretical insights into a structured view of higher category theory. The results obtained hold potential for profound implications in both theoretical foundations and broader applications, notably in algebraic topology and related areas.

While the paper is grounded in deep theoretical constructs, its design is to inspire future developments where further expanding or relaxing axiomatic structures could lead to new revelations or applications across mathematical disciplines. As of its publication, it sets a high bar for rigor and clarity in tackling the complexities involved in higher-dimensional category theoretical constructs. Future work in this area might explore relaxation of some axioms in specific contexts or the implications of such a universal characterization in computational or applied mathematics.