nLab cobordism category

Idea

The notion of cobordism categories in the original sense of Stong 1968 abstracts basic properties of (variants of) categories whose objects are compact manifolds with boundary, with the intent of regarding these as cobordisms between their boundary components.

A closely related but nominally different notion are categories whose morphisms are taken to be cobordisms between their boundary components.

Beware that the use of terminology not always brings out this distinction; but these days the second meaning is more prevalent, in particular in discussion of cobordism cohomology and of topological field theory.

Definitions

Cobordisms as objects

The axiomatization below is motivated as capturing the following familiar situation:

The category $D$ of compact smooth manifolds with boundary, has finite coproducts and the boundary operator $\partial \colon D\to D$, $M\mapsto \partial M$ is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums. Notice that $D$ is similar to but not actually an additive category. The inclusions $i_M \colon \partial M\to M$ form a natural transformation of functors $i \colon \partial\to Id$. Finally, the isomorphism classes of objects in $D$ form a set, so $D$ is essentially small (svelte).

Definition

A Stong cobordism category is a triple $(D,\partial,i)$ where

• $D$ is a svelte category (i.e. an essentially small category)

  • with finite coproducts (called direct sums, often denoted by $+$),

  • including an initial object $0$ (also often denoted by $\emptyset$),

• $\partial:D\to D$ is an additive (direct-sum-preserving) functor

• and $i:\partial\to Id_D$ is a natural transformation such that $\partial\partial M = 0$ for all objects $M\in D$.

Stong 1968, p. 4.

Remark

Note that $i$ is not required to be a subfunctor of the identity, i.e. the components $i_M$ are not required to be monic, which is however often the case in examples.

Definition

Two objects $M$ and $N$ in a cobordism category $(D,\partial,i)$ are said to be cobordant, written $M\sim_{cob} N$, if there are objects $U,V\in D$ such that $M+\partial U \cong N+\partial V$ where $\cong$ denotes the relation of being isomorphic in $D$.

Remark

In particular, isomorphic objects are cobordant. Being cobordant is an equivalence relation and for any object $M$ in $D$, one has $\partial M\sim_{cob} 0$.

Definition

Objects of the form $\partial M$ where $M$ is an object in $D$ are said to be boundaries and the objects $V$ such that $\partial V = 0$ are said to be closed.

Remark

In particular, every boundary is closed. A direct sum of closed objects (resp. boundaries) is a closed object (resp. a boundary). If an object $M$ is a boundary and $M\cong N$ then $N$ is also a boundary.

Definition

By the above, the relation of being cobordant is compatible with the direct sum, in the sense that the direct sum induces an associative commutative operation on the set of equivalence classes, which hence becomes a commutative monoid called the cobordism semigroup

$$\Omega(D,\partial,i) \,,$$

of the cobordism category $(D,\partial,i)$.

Cobordisms as morphisms

(…)

e.g. GMWT09, 2.1

(…)

Properties

The following properties concern the notion ob cobordism categories with cobordisms serving as morphisms.

The homotopy type of the cobordism category

Topological case

Theorem

There is a weak homotopy equivalence

$$\Omega |Cob_d| \simeq \Omega^\infty(MTSO(d))$$

between the loop space of the geometric realization of the $d$-cobordism category and the Thom spectrum-kind spectrum

$$\Omega^\infty MTSO(d) := {\lim_\to}_{n \to \infty} \Omega^{n+d} Th(U_{d,n}^\perp)$$

where

$$U_{d,n}^\perp = \{ ... \}$$

This is Galatius, Tillmann, Madsen & Weiss 2009, main theorem.

Remark

This statement may be thought of as a limiting case, of the cobordism hypothesis-theorem. See there for more.

Geometric case

• (Ayala)

The Thom group? $\mathcal{N}_*$ of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for $D=Diff_c$.

Examples

Examples of cobordism categories besides those of manifolds with $(B,f)$-structure:

• “foams” used in link homology following Khovanov 2004

See also MO:q/59677.

Related concepts

• cobordism, extended cobordism

• category of cobordisms

  • cobordism ring

  • Thom spectrum

  • (∞,n)-category of cobordisms

• conformal cobordism category

• cobordism hypothesis

• cobordism cohomology theory

References

General

The notion of cobordism categories with cobordisms as objects is due to

• Robert Stong, Notes on cobordism theory, Princeton University Press 1968 (Russian transl., Mir 1973) [toc pdf, ISBN:9780691649016]

Most authors these days use “cobordism category” to refer to the notion where cobordisms form the morphisms:

The GMTW theorem about the homotopy type of the cobordisms category with topological structures on the cobordisms appears in

• Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss, The homotopy type of the cobordism category, Acta Math. 202 2 (2009) 195–239 [arXiv:math/0605249]

A generalization to geometric structure on the cobordisms is discussed in

• David Ayala, Geometric cobordism categories thesis (2009) (arXiv:0811.2280)

Embedded cobordism category

• Oscar Randal-Williams, Embedded Cobordism Categories and Spaces of Manifolds, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)

On the homotopy groups of the embedded cobordism category:

• Marcel Bökstedt, Anne Marie Svane, A geometric interpretation of the homotopy groups of the cobordism category, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)

• Marcel Bökstedt, Johan Dupont, Anne Marie Svane, Cobordism obstructions to independent vector fields, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)