under construction
$n$-Dimensional manifolds (possibly and usually equipped with certain structure, notably for instance with orientation, framing-structure or more general G-structure) 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 $n$-dimensional cobordisms;
(n+2)-morphisms are smooth homotopies of these;
etc.
The $(\infinity,n)$-category of cobordisms is the subject of the cobordism hypothesis.
Here is an outline of the idea of the definition of $Bord_{(\infty,n)}$ as given in (Lurie) 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 $n$-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 $n$-dimensional cobordisms.
consider a manifold $X \hookrightarrow V \times \mathbb{R}$ embedded in a vector space of the form $V \times \mathbb{R}$. We can think of this as a manifold canonically equipped with a coordinate function $\phi : X \hookrightarrow V \times \mathbb{R} \to \mathbb{R}$ that measures the “height” or maybe better the “length” of the embedded manifold.
We can pick a bunch of numbers $\{t_j \in \mathbb{R}\}$ and think of these as marking a bunch of slices of $X$, the preimages $\phi^{-1}(t_j)$. We can think of these slices as being the $(n-1)$-dimensional boundary manifolds at which a sequence of manifolds have been glued together to produce $X$.
(there is an obvious picture to be drawn and uploaded here, maybe somebody finds the time and energy)
In this way an embedded manifold $X \hookrightarrow V \times \mathbb{R}$ and a set of $k$-numbers $\{t_i\}$ may represent an element in the space of sequences of composable cobordisms. To make this work as expected, the markings on $X$ 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 $X \hookrightarrow V \times \mathbb{R}$ and a collection of $k$ numbers $\{t_i \in \mathbb{R}\}$ naturally form a simplicial set, which is like the nerve of the 1-category of $n$-dimensional cobordisms under composition in one direction.
To generalize this from just a 1-categorical structure to an $n$-categorical structure, we simply take a manifold $X$ as before, but now draw markings on it in $n$ 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 $n$-dimensional pasting diagrams in an $n$-fold category.
To formalize this more general case, we embed $X$ not just into a $V \times \mathbb{R}$, but a $V \times \mathbb{R}^n$. This then provides us with $n$ different coordinate functions $\phi_i : X \hookrightarrow V \times \mathbb{R}^n \stackrel{p_i}{\to} \mathbb{R}$ on $X$, each running along one of the directions in which we may think of $X$ as having been glued from smaller manifolds.
A collection of markings indicating such gluing is now a collection of numbers $\{t_j^1\}, \;\{t_j^2\}, \; \cdots \{t_j^n\}$, 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 $n$-fold simplicial set
which is like the nerve of an $n$-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 $n$-fold simplicial topological space
To get rid of the dependence of this construction on $V$, we can let $V$ “grow arbitrarily large” by taking the colimit of the above $n$-fold cosimplicial spaces as $V$ ranges over the finite dimensional subspaces of $\mathbb{R}^\infty$.
The resulting $n$-fold simplicial topological space obtained by this colimit then is essentially the (∞,n)-category $Bord_n$ that we are after. It turns out that it actually is an $n$-fold Segal space. We just formally complete it to an n-fold complete Segal space
This, then, is a model for the (∞,n)-category of extended $n$-dimensional cobordisms.
There is a definition of a blob n-category of $n$-cobordisms. See there for more details.
Some comments on 2-framed 2-cobordisms.
Consider the pictures in (Schommer-Pries 13, figure 5).
Somebody should produce pictures like this here…
Well, it worked in the preview; a little help?
Let $\gamma$ be a 1-dimensional manifold of the form of the interval $[0,1]$. A 2-framing of $\gamma$ is a trivialization of $T\gamma \oplus \mathbb{R}$. Let $\{1\} \subset \mathbb{R}$ be the canonical basis of $\mathbb{R}$. If we think of the plane $\mathbb{R}^2$ as equipped with its canonical 2-framing, then a 2-framing of $\gamma$ is induced by embedding $\gamma$ into the plane and shading one of its two sides. This identifies at each point $x \in \gamma$ the tangent space to $\gamma$ at that point with the tangent vector to the embedding of $\gamma$ as a vector in $\mathbb{R}^2$ and identifies $1\in \mathbb{R}$ with the vector in $\mathbb{R}^2$ orthogonal to this tangent vector and pointing into the shaded region.
This shows that if $\gamma$ is regarded with its two endpoints both as incoming or both as outgoing, then the induced 2-framing of these endpoints is opposite to each other. This way such an arc is a morphism from the union of the “positive point” and the “negative point” to the empty 0-manifold, hence is a unit/counit exhibiting these as dual objects.
$Bord_n$ is an (∞,n)-category with all adjoints.
For $n \to \infty$ we have that $Bord_{(\infty,\infty)}$ is the symmetric monoidal ∞-groupoid ($\simeq$ infinite loop space) $\Omega^\infty M O$ that underlies the Thom spectrum.
Its homotopy groups are the cobordism rings
Therefore a symmetric monoidal $\infty$-functor
to some symmetric monoidal $\infty$-groupoid $S$ is a genus.
(∞,n)-category of cobordisms
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
Other discussions of higher categories of cobordisms are
Eugenia Cheng and Nick Gurski, Toward an $n$-category of cobordisms , Theory and Applications of Categories 18 (2007), 274-302. (tac)
A detailed construction of the (2,2)-category of 2-dimensional cobordisms is
Chris Schommer-Pries, 2-category of 2-dimensional cobordisms.
Chris Schommer-Pries, Dualizability in Low-Dimensional Higher Category Theory (arXiv:1308.3574)
For a discussion of the relation of $Bord_{(\infty,\infty)}$ to the Thom spectrum and the cobordism ring see also