nLab
(infinity,n)-category of cobordisms

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Manifolds and Cobordisms

Functorial Quantum Field Theory

Contents

Idea

nn-Dimensional manifolds (possibly and usually equipped with certain structure, notably for instance with orientation) should naturally form an (∞,n)-category of extended cobordisms whose

  • objects are 0-dimensional (oriented) manifolds (disjoint unions of (oriented) points);

  • 1-morphisms are (oriented) cobordisms between disjoint unions of (oriented) points;

  • 2-morphisms are cobordisms between 1-dimensional cobordisms

  • etc.

  • (n+1)-morphisms are diffeomorphisms between nn-dimensional cobordisms;

  • (n+2)-morphisms are smooth homotopies of these;

  • etc.

The (,n)(\infinity,n)-category of cobordisms is the subject of the cobordism hypothesis.

Definition

As an nn-fold complete Segal space

Here is an outline of the idea of the definition of Bord (,n)Bord_{(\infty,n)} as given in

where the main point, apart from the (∞,n)-category machinery in the background, is definition 2.2.9.

The idea is to start with thinking of nn-dimensional cobordisms as forming something like an n-fold category by simply saying that the collection of composites of cobordisms is given by big cobordisms with markings on them, indicating where we think of them as being composed.

Let’s first do this for composition in one direction, as in an ordinary 1-category of nn-dimensional cobordisms.

consider a manifold XV×X \hookrightarrow V \times \mathbb{R} embedded in a vector space of the form V×V \times \mathbb{R}. We can think of this as a manifold canonically equipped with a coordinate function ϕ:XV×\phi : X \hookrightarrow V \times \mathbb{R} \to \mathbb{R} that measures the “height” or maybe better the “length” of the embedded manifold.

We can pick a bunch of numbers {t j}\{t_j \in \mathbb{R}\} and think of these as marking a bunch of slices of XX, the preimages ϕ 1(t j)\phi^{-1}(t_j). We can think of these slices as being the (n1)(n-1)-dimensional boundary manifolds at which a sequence of manifolds have been glued together to produce XX.

(there is an obvious picture to be drawn and uploaded here, maybe somebody finds the time and energy)

In this way an embedded manifold XV×X \hookrightarrow V \times \mathbb{R} and a set of kk-numbers {t i}\{t_i\} may represent an element in the space of sequences of composable cobordisms. To make this work as expected, the markings on XX may not be too irregular, so we should impose some conditions on what qualifies as a marked manifold. The precise statement is given further below.

The collection of these tuples, consisting of an embedded manifold XV×X \hookrightarrow V \times \mathbb{R} and a collection of kk numbers {t i}\{t_i \in \mathbb{R}\} naturally form a simplicial set, which is like the nerve of the 1-category of nn-dimensional cobordisms under composition in one direction.

To generalize this from just a 1-categorical structure to an nn-categorical structure, we simply take a manifold XX as before, but now draw markings on it in nn transversal directions, thereby putting a kind of grid on it that subdivides the manifold into cubical slices. A manifold with such subdivision on it may then be regarded as giving an element in the space of nn-dimensional pasting diagrams in an nn-fold category.

To formalize this more general case, we embed XX not just into a V×V \times \mathbb{R}, but a V× nV \times \mathbb{R}^n. This then provides us with nn different coordinate functions ϕ i:XV× np i\phi_i : X \hookrightarrow V \times \mathbb{R}^n \stackrel{p_i}{\to} \mathbb{R} on XX, each running along one of the directions in which we may think of XX as having been glued from smaller manifolds.

A collection of markings indicating such gluing is now a collection of numbers {t j 1},{t j 2},{t j n}\{t_j^1\}, \;\{t_j^2\}, \; \cdots \{t_j^n\}, one for each of these directions.

For each direction this yields a simplicial set of such structures, to be thought of as the nerve of the category of cobordisms under composition in one of these directions. Taken together this is an nn-fold simplicial set

Δ op×Δ op××Δ opSet \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Set

which is like the nerve of an nn-fold category of cobordisms.

When suitable regularity conditions are imposed on this data, there is naturally a topology on each of these sets of embedded marked cobordisms, that makes this into an nn-fold simplicial topological space

Δ op×Δ op××Δ opTop. \Delta^{op} \times \Delta^{op} \times \cdots \times \Delta^{op} \to Top \,.

To get rid of the dependence of this construction on VV, we can let VV “grow arbitrarily large” by taking the colimit of the above nn-fold cosimplicial spaces as VV ranges over the finite dimensional subspaces of \mathbb{R}^\infty.

The resulting nn-fold simplicial topological space obtained by this colimit then is essentially the (∞,n)-category Bord nBord_n that we are after. It turns out that it actually is an nn-fold Segal space. We just formally complete it to an n-fold complete Segal space

Bord n:(Δ op) nTop. Bord_n : (\Delta^{op})^n \to Top \,.

This, then, is a model for the (∞,n)-category of extended nn-dimensional cobordisms.

As a blob nn-category

There is a definition of a blob n-category of nn-cobordisms. See there for more details.

Properties

Adjoints

Bord nBord_n is an (∞,n)-category with all adjoints.

Relation to Thom spectrum

For nn \to \infty we have that Bord (,)Bord_{(\infty,\infty)} is the symmetric monoidal ∞-groupoid (\simeq infinite loop space) Ω MO\Omega^\infty M O that underlies the Thom spectrum.

Its homotopy groups are the cobordism rings

π nBord (,)Ω n. \pi_n Bord_{(\infty,\infty)} \simeq \Omega_n \,.

Therefore a symmetric monoidal \infty-functor

Bord (,)S Bord_{(\infty,\infty)} \to S

to some symmetric monoidal \infty-groupoid SS is a genus.

References

A specific realization of this idea in terms of (∞,n)-category modeled as n-fold complete Segal space is in (definition 2.2.9, page 36)

In that article a proof of the cobordism hypothesis is indicated. A review is in

A detailed construction of the (2,2)-category of 2-dimensional cobordisms is

For a discussion of the relation of Bord (,)Bord_{(\infty,\infty)} to the Thom spectrum and the cobordism ring see also

Other discussions of higher categories of cobordisms are

Revised on April 27, 2013 18:29:02 by Urs Schreiber (89.204.138.212)