category with duals (list of them)
dualizable object (what they have)
with induced orientation agreeing with the given one on and being the opposite of that of . Hence by labelling disjoint components of the boundary of any manifold with boundary as either “incoming” or “outgoing”, then becomes a cobordism from its incoming to its outgoing boundary components.
(While is the boundary of , conversely is the “co-boundary” of . This is part of the reason for the “co-” in “cobordism”, but sometimes one just says bordism.)
With a little bit of technical conditions on the boundary inclusions added, then -dimensional manifolds with diffeomorphism classes of -dimensional cobordisms between them form a category (a category of cobordisms) whose objects are the manifolds, whose morphisms are the cobordisms, and whose composition operation is the operation of gluing two cobordisms along a common boundary component. Such a category of cobordisms of some dimension is naturally a symmetric monoidal category with the tensor product being the disjoint union of manifolds.
The connected components in this category are called cobordism classes of manifolds. Under cartesian product and disjoint union, these form what is called the cobordism ring in the given dimension. The study of these classes, hence of manifolds “up to cobordisms”, is a central topic in algebraic topology.
A central insight connecting algebraic topology with mathematical physics is that a strong monoidal functor on a category of cobordisms with values in something like the category Vect (with its standard tensor product of modules) may be thought of as a formalized incarnation of what in physics is called a topological quantum field theory
Here one thinks of a cobordism as a piece of spacetime (or worldvolume) of dimension , and of the -dimensional manifolds that this goes between as a piece of space (or brane). A functor as above is then thought of as sending each -dimensional space to its space of quantum states and each spacetime between and to a linear map .
From the perspective of mathematics, such a functor is a way to break up diffeomorphism invariants of closed manifolds of dimension into pieces and being able to reconstruct them from gluing of data associated to manifolds with -dimensional boundary.
If one considers manifolds with corners then there is a fairly evident refinement of the concept of cobordism that allows to further refine this process to gluing of extended cobordisms of any dimension going between extended cobordisms of dimension . Such extended cobordisms of maximal dimension form a symmetric monoidal (infinity,n)-category called, naturally, the (infinity,n)-category of cobordisms. The cobordism hypothesis asserts that this is a most fundamental object in higher category theory and higher algebra, namely that it is the free symmetric monoidal (infinity,n)-category with duals. The corresponding extended concept of topological quantum field theory is accordingly called extended TQFT or similar.