FQFT and cohomology
The notion of cobordism category is an abstract one intended to capture important features of (many variants of) the category of cobordisms and include in the same formalism cobordisms for closed manifolds with various kinds of structure.
The passage from a manifold to its boundary has some formal properties which are preserved in the presence of orientation, for manifolds with additional structure and so on. The category of compact smooth manifolds with boundary has finite coproducts and the boundary operator , is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums, and some say that is an additive functor, but is not actually an additive category). The inclusions form a natural transformation of functors . Finally, the isomorphism classes of objects in form a set, so is essentially small (svelte).
A cobordism category is a triple where
Two objects and in a cobordism category are said to be cobordant, written , if there are objects such that where denotes the relation of being isomorphic in .
In particular, isomorphic objects are cobordant. Being cobordant is an equivalence relation and for any object in , one has .
Objects of the form where is an object in are said to be boundaries and the objects such that are said to be closed.
In particular, every boundary is closed. A direct sum of closed objects (resp. boundaries) is a closed object (resp. a boundary). If an object is a boundary and then is also a boundary.
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
of the cobordism category .
There is a weak homotopy equivalence
This is (GalatiusTillmannMadsenWeiss, main theorem).
The Thom group? of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for .
category of cobordisms
A classical reference is
A generalization to geometric structure on the cobordisms is discussed in