Categorical Aspects
Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a category whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of and is defined by gluing the right end of the first to the left end of the second, yielding . A cobordism is a kind of cospan: M → W ← N.
A topological quantum field theory is a monoidal functor from a category of cobordisms to a category of vector spaces. That is, it is a functor whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
In low dimensions, the bordism question is trivial, but the category of cobordism is still interesting. For instance, the disk bounding the circle corresponds to a null-ary operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
Read more about this topic: Cobordism
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