Cobordism categories that don't involve manifolds

Question

In order to capture the various flavors of cobordism into one concept, the notion of a "cobordism category" is introduced. This is an essentially small category $C$, together with finite coproducts, an initial object, and an additive functor $\partial$ satisfying some properties. Of course, the idea is that one should think of the objects of $C$ as manifolds of one sort or another and $\partial$ as taking the boundary. Indeed, in Tom Weston's notes on the subject he immediately restricts his attention to cobordism categories of "$(B,f)$" manifolds, which are a special type of manifold.

The question is: What are some examples of cobordism categories $(C, \partial, i)$, where the objects of $C$ are not manifolds of some kind?

The only example I know of is motivic cobordism. I'm hoping there are others!

(For the reader's convenience, here is the definition:

A cobordism category is an essentially small category $C$ with finite coproducts (including an initial object $0$), equipped with a coproduct-preserving functor $\partial: C \to C$ and a natural transformation $i: \partial \to 1_C$, such that $\partial\partial c \cong 0$ for every object $c$.)

Answer 1

There is actually a reasonable amount of literature on the subject.

Check out Khovanov's $sl_3$-homology and the subsequent work by Marco Mackaay and Pedro Vaz * * *, Hao Wu, and Scott Morrison and Ari Nieh. The objects are webs, the morphisms are foams. The webs have trivalent vertices and foams are modeled on the dual complex of a tetrahedron. I also liked a paper by Serge Natanzon on Network TQFT. You an find all these papers on the Arxiv.

Early on, Frank Quinn made some forays into TQFT for $2$-complexes with the idea of detecting counterexamples to the Andrews Curtis conjecture.

There are older papers of John Baez, John Barret, and then lots of papers in the physical literature. The Landau-Ginzberg approach to quantum gravity involves webs and foams.

All of this work can be understood as being about $3+1$-dimensional TQFT.

In the $1+1$ case, TQFT's turned out to be Frobenius algebras in disguise. In the $2+1$ case, TQFT's have to do with spherical categories. The corresponding correspondence for $3+1$ has yet to be understood. I would conjecture that on the way to understanding $3+1$-dimensional TQFT, a suitable and uniformly accepted theory of TQFT based on webs and foams will be established.

Answer 2

Here is one type of example, basically inspired by the manifold-type examples but so general that they are not actually categories of manifolds. Let $B$ be any small category with finite coproducts, and let $C$ be the category of diagrams of shape

$$X \to Z \leftarrow Y$$

in $B$. Coproducts in $C$ are given pointwise by coproducts in $B$. Define

$$\partial(X \to Z \leftarrow Y) = (0 \to X + Y \leftarrow 0)$$

with the obvious extension to morphisms. It is quite clear that $\partial$ preserves coproducts and that $\partial^2 \cong 0$. Also there is a canonical natural transformation $i: \partial \to 1_C$, whose component at the object $X \to Z \leftarrow Y$ is the unique one where the arrow in the middle is the map $X + Y \to Z$ whose restrictions to $X$ and $Y$ are the given arrows $X \to Z$, $Y \to Z$ of the object.

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Edit: As indicated in a comment below, it is simpler to consider instead the arrow category $B^{\mathbf{2}}$ as a cobordism category where $\partial(f: X \to Y) = (0 \to X)$. But a much more compelling reason to consider this construction is that, if I'm not mistaken, it satisfies a universal property as follows.

Let $\text{Coprod}$ be the 2-category of categories with finite coproducts and coproduct-preserving functors (and transformations between them); let $\text{Cobord}$ be the 2-category of cobordism categories and cobordism-preserving functors. There is an evident forgetful 2-functor

$$U: \text{Cobord} \to \text{Coprod}$$

In the other direction, the arrow-cobordism category construction defines a 2-functor

$$\text{Arr}: \text{Coprod} \to \text{Cobord}$$

and this is in fact a right 2-adjoint of the forgetful functor. (Notation: $U \dashv \text{Arr}$.) Thus $\text{Arr}(B) = B^{\mathbf{2}}$ defines the cofree cobordism category generated by a category with coproducts $B$.

In more detail, the unit $\eta: 1_{\text{Cobord}} \to \text{Arr} \circ U$ is defined componentwise as a cobordism-preserving functor $\eta C: C \to C^{\mathbf{2}}$ which at the object level takes an object $c$ to the object $i c: \partial c \to c$ in $C^{\mathbf{2}}$. (It is instructive to check the details of this.) The counit $\varepsilon: U \circ \text{Arr} \to 1_{\text{Coprod}}$ is defined componentwise as a coproduct-preserving functor $\varepsilon D: D^{\mathbf{2}} \to D$ which at the object level takes an object $g: d_1 \to d_2$ to the object $d_2$. It is reasonably straightforward to check that there are coherent isomorphisms

$$(U \stackrel{U \eta}{\to} U \circ \text{Arr} \circ U \stackrel{\varepsilon U}{\to} U) \cong 1_U$$

$$(\text{Arr} \stackrel{\eta \text{Arr}}{\to} \text{Arr} \circ U \circ \text{Arr} \stackrel{\text{Arr} \varepsilon}{\to} \text{Arr}) \cong 1_{\text{Arr}}$$

that make $\text{Arr}$ the right 2-adjoint of the forgetful functor $U$.

I think there's something deeper going on here than I understand at the present time.

Answer 3

One famous (in my field!) example is Witt space bordism. Witt spaces are not manifolds but rather pseudomanifolds (which aren't so far off from manifolds, but they can have singularities). A pseudomanifold is a Witt space if certain local rational intersection homology groups vanish. The bordism group of Witt spaces is important because it turns out to be a geometric model for ko-homology after inverting 2. The original reference is Paul Siegel's thesis: http://www.jstor.org/stable/2374334 There's a slightly fancier version due to Pardon using "IP spaces" which satisfy integral Poincare duality: http://www.springerlink.com/content/6m5j386lr5hx2444/