A note on the (∞,n)–category of cobordisms

@article{Calaque2019ANO,
  title={A note on the (∞,n)–category of cobordisms},
  author={Damien Calaque and Claudia I. Scheimbauer},
  journal={Algebraic \& Geometric Topology},
  year={2019}
}
In this note we give a precise definition of fully extended topological field theories a la Lurie. Using complete n-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms and the cobordism bicategory from it. 

Two-Dimensional Extended Homotopy Field Theories

We define $2$-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When target is a $K(G,1)$-space, oriented E-HFTs taking values in the symmetric monoidal

A T ] 1 N ov 2 02 1 The geometric cobordism hypothesis

  • Mathematics
  • 2021
We prove a generalization of the cobordism hypothesis of Baez–Dolan and Hopkins–Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures,

Equivariant higher Hochschild homology and topological field theories

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group $G$. As coefficients, we allow $E_\infty$-algebras with $G$-action. For this

The geometric cobordism hypothesis

We prove a generalization of the cobordism hypothesis of Baez–Dolan and Hopkins–Lurie for bordisms with arbitrary geometric structures, such as Riemannian or Lorentzian metrics, complex and

Cartesian Fibrations of Complete Segal Spaces

Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories. In this work

An Introduction to Higher Categories

  • S. Paoli
  • Mathematics
    Algebra and Applications
  • 2019
In this chapter we give a non-technical introduction to higher categories. We describe some of the contexts that inspired and motivated their development, explaining the idea of higher categories,

Tori detect invertibility of topological field theories

A once-extended d-dimensional topological field theory Z is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal (infty,2)-category) assigning values to (d-2)-manifolds,

On straightening for Segal spaces

We study cocartesian fibrations between (∞, d)-categories in terms of higher Segal spaces and prove a version of straightening for them. This follows from a repeated application of an explicit

Quasi-categories vs. Segal spaces: Cartesian edition

  • Nima Rasekh
  • Mathematics
    Journal of Homotopy and Related Structures
  • 2021
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On

2–Segal objects and the Waldhausen construction

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we

References

SHOWING 1-10 OF 61 REFERENCES

A cartesian presentation of weak n–categories

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of

A Survey of (∞, 1)-Categories

In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all

The homotopy type of the cobordism category

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main

Rigidification of higher categorical structures

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of "up to homotopy" models for this limit sketch in a suitable model category can be transferred

From fractions to complete Segal spaces

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This

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

In this paper we introduce the models for (∞, 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 (∞,

Iterated spans and classical topological field theories

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of higher-categorical local systems, and classify their fully dualizable objects. By the Cobordism

A MODEL STRUCTURE ON INTERNAL CATEGORIES IN SIMPLICIAL SETS

We put a model structure on the category of categories internal to sim- plicial sets. The weak equivalences in this model structure are preserved and reected by the nerve functor to bisimplicial sets

Comparison of models for $(\infty, n)$-categories, II

In this paper we complete a chain of explicit Quillen equivalences between the model category for $\Theta_{n+1}$-spaces and the model category of small categories enriched in $\Theta_n$-spaces.

ON COBORDISM OF MANIFOLDS WITH CORNERS

This work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of
...