Cobordism category - Imported Knowlpedia Content

Definition

For $d\geq1$ , the $d$ -dimensional cobordism category $\mathbf{Cob}_d$ has closed smooth manifolds of dimension $d-1$ as objects. A morphism $M_0\to M_1$ is a diffeomorphism class, relative to the boundary, of compact $d$ -dimensional cobordisms $W$ equipped with boundary identifications $\partial W\cong M_0\sqcup M_1$ and collars of those boundary components. Composition is smooth gluing along the identified common boundary, disjoint union is the symmetric monoidal product, and the cylinder represents the identity morphism. All diffeomorphisms used in the quotient must preserve the specified boundary data.

Why collars enter

A collar identifies a neighborhood of each boundary component with a product. Product coordinates make two cobordisms glue to a smooth manifold and show that different compatible collar choices produce diffeomorphic composites. Passing to diffeomorphism classes removes the remaining choices and makes composition associative. Keeping embeddings and collars instead leads naturally to a topological category.

Structured variants

The oriented cobordism category uses oriented objects and oriented cobordisms, with incoming boundary orientation reversed. Tangential structures give framed, spin, complex, and other bordism categories. One may also require maps to a fixed background space. In every case, composition is permitted only when the structure on the common boundary matches and extends across the glued manifold.

Topological enrichment

The category used in homotopy theory often retains spaces of embedded manifolds and cobordisms rather than collapsing morphisms to diffeomorphism classes. Galatius–Madsen–Tillmann–Weiss define such a topological category and identify the weak homotopy type of its classifying space with an infinite loop space associated to a Thom spectrum GMTW, §2 and Main Theorem. This enriched category and the ordinary quotient category encode related but different information.

Relation to field theory

A $d$ -dimensional topological quantum field theory in the Atiyah–Segal sense is a symmetric monoidal functor from an appropriately structured cobordism category to a category such as vector spaces. Gluing cobordisms becomes composition of linear maps, while disjoint union becomes tensor product. The categorical formulation makes locality under cutting and gluing an algebraic axiom Baez–Dolan, §1.

References

Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss, “The homotopy type of the cobordism category,” Acta Mathematica 202 (2009), 195–239. DOI record. Relevant: §2 and the main theorem.
John C. Baez and James Dolan, “Higher-dimensional algebra and topological quantum field theory,” in Category Theory, Contemporary Mathematics 230, AMS, 1998. DOI record. Relevant: §1, cobordisms and symmetric monoidal functors.