Models for $(infty, n)$-categories and the cobordism hypothesis

In this paper we introduce the models for $(\infty, n)$-categories which have been developed to date, as well as the comparisons between them that are known and conjectured. We review the role of $(\infty, n)$-categories in the proof of the Cobordism Hypothesis.

💡 Research Summary

The paper provides a panoramic overview of the landscape of models for $(\infty,n)$‑categories, surveys the known and conjectural equivalences between them, and explains how these structures underpin the proof of the Cobordism Hypothesis. It begins by recalling the definition of an $(\infty,n)$‑category: a higher‑category in which all $k$‑morphisms for $k>n$ are invertible, while morphisms up to level $n$ may be non‑invertible. This notion has become indispensable in modern topology, algebraic geometry, and quantum field theory because it simultaneously encodes homotopical data and categorical composition at several layers.

The authors then enumerate the principal models that have been constructed to date. The first family consists of $n$‑fold complete Segal spaces, obtained by iterating Rezk’s complete Segal space construction $n$ times. Each direction carries its own Segal condition and completeness requirement, yielding a model structure that is simultaneously a Reedy model in each simplicial direction and a left Bousfield localization enforcing the Segal maps. The second family is $\Theta_n$‑spaces, introduced by Joyal and Lurie. Here the indexing category $\Theta_n$—a higher‑dimensional analogue of the simplex category—provides a combinatorial skeleton for cells of all dimensions, and presheaves on $\Theta_n$ satisfying Segal and completeness axioms model $(\infty,n)$‑categories. The third approach uses $n$‑fold simplicial categories (or $n$‑fold simplicial objects in Cat), where each simplicial direction records a hom‑space enriched in simplicial sets; fibrant objects are precisely those whose multi‑simplicial enrichment satisfies a homotopy coherent composition law. The fourth line of development is based on relative categories equipped with a marked model structure (the “marked” or “Cartesian” model). By specifying a class of weak equivalences one obtains a homotopical category that can be localized to an $(\infty,1)$‑category, and then iterated to reach the $(\infty,n)$ level. Finally, the operadic models—most notably $E_n$‑algebras and their higher‑dimensional analogues such as $n$‑fold Segal operads—capture the symmetric monoidal structure and the dualizability conditions required for the Cobordism Hypothesis.

Having listed the models, the paper turns to the web of comparisons. The most robust results are Quillen equivalences. Barwick and Schommer‑Pries proved a chain of Quillen equivalences linking $n$‑fold complete Segal spaces to $\Theta_n$‑spaces, showing that both present the same underlying $(\infty,n)$‑category. Bergner and Rezk constructed a pair of adjoint functors between $n$‑fold simplicial categories and $\Theta_n$‑spaces, establishing a Quillen equivalence after appropriate left Bousfield localizations. Lurie and Hopkins demonstrated that the operadic model of $E_n$‑algebras is Quillen equivalent to the $\Theta_n$‑space model when both are equipped with symmetric monoidal structures. However, several conjectural equivalences remain open. The most prominent is the “model independence conjecture”: every known model should be Quillen equivalent to any other, implying a unique homotopy theory of $(\infty,n)$‑categories. In particular, a direct Quillen equivalence between the relative‑category model and the operadic model has not yet been constructed, nor has a fully explicit comparison between $n$‑fold complete Segal spaces and $n$‑fold simplicial categories been published. These gaps motivate ongoing research.

The second half of the paper is devoted to the Cobordism Hypothesis. In its modern formulation (due to Baez–Dolan and proved by Lurie), the hypothesis asserts that for any symmetric monoidal $(\infty,n)$‑category $\mathcal{C}$, the space of fully extended $n$‑dimensional topological quantum field theories (TQFTs) with target $\mathcal{C}$ is equivalent to the space of fully dualizable objects of $\mathcal{C}$. The authors explain how each model supplies a concrete realization of the source $(\infty,n)$‑category $\mathrm{Bord}_n$, whose objects are closed $(n-1)$‑manifolds, 1‑morphisms are $n$‑dimensional cobordisms, and higher morphisms are diffeomorphisms, isotopies, and higher isotopies. Using the $\Theta_n$‑space model, one can define $\mathrm{Bord}_n$ as a presheaf on $\Theta_n$ satisfying Segal and completeness conditions; the Segal maps encode gluing of cobordisms, while completeness guarantees the existence of units. The operadic model shines when discussing dualizability: an object $X\in\mathcal{C}$ is fully dualizable if it admits left and right adjoints at every level $k\le n$, together with evaluation and coevaluation $k$‑morphisms satisfying the zig‑zag identities. These data are precisely the algebraic structure of an $E_n$‑algebra in $\mathcal{C}$, making the operadic viewpoint natural for the hypothesis. The paper walks through Lurie’s proof strategy: first construct a symmetric monoidal functor $\mathrm{Bord}_n\to\mathcal{C}$ from a fully dualizable object, then show that any such functor is uniquely determined up to contractible choice. The proof relies heavily on the existence of the aforementioned Quillen equivalences, because one needs to move between a model where composition is easy to describe (e.g., $\Theta_n$‑spaces) and a model where the monoidal structure and duals are transparent (e.g., operadic $E_n$‑algebras).

In the concluding section the authors outline future directions. They stress the importance of resolving the remaining comparison conjectures, which would give a fully model‑independent foundation for $(\infty,n)$‑category theory. They also advocate the development of computational tools—such as explicit $\Theta_n$‑cell complexes or software implementations of $n$‑fold Segal spaces—that could make the classification of extended TQFTs more concrete. Finally, they point to the potential of extending the Cobordism Hypothesis beyond topological field theories, for instance to conformal or holomorphic settings, where the interplay between higher categories and geometric structures is still largely unexplored.

Overall, the paper serves as both a reference guide to the current zoo of $(\infty,n)$‑category models and a roadmap for how these models collectively enable the deep and far‑reaching Cobordism Hypothesis. By clarifying the known equivalences and highlighting the open conjectures, it lays the groundwork for a unified, computationally tractable theory of fully extended topological quantum field theories.