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 $c$.)