cobordism category (Rev #2) in nLab

Idea

The notion of cobordism category is an abstract one intended to capture important features of the category of cobordisms.

Motivation

The passage from a manifold MM to its boundary ∂M\partial M 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 D=Diff cD = Diff_c has finite coproducts and the boundary operator ∂:D→D\partial:D\to D, M↦∂MM\mapsto \partial M is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums, and some say that ∂\partial is an additive functor, but DD is not actually an additive category). The inclusions i M:∂M→Mi_M:\partial M\to M form a natural transformation of functors i:∂→Idi:\partial\to Id. Finally, the isomorphism classes of objects in DD form a set, so DD is essentially small (svelte).

Definition

A cobordism category is a triple (D,∂,i)(D,\partial,i) where DD is a svelte category with finite coproducts (called direct sums, often denoted by ++), including an initial object 00 (also often denoted by ∅\emptyset), ∂:D→D\partial:D\to D is an additive (direct-sum-preserving) functor and i:∂→Id Di:\partial\to Id_D is a natural transformation such that ∂∂M=0\partial\partial M = 0 for all objects M∈DM\in D.

Note that ii is not required to be a subfunctor of the identity, i.e. the components i Mi_M are not required to be monic, which is however often the case in examples.

Two objects MM and NN in a cobordism category (D,∂,i)(D,\partial,i) are said to be cobordant, written M∼ cobNM\sim_{cob} N, if there are objects U,V∈DU,V\in D such that M+∂U≅N+∂VM+\partial U \cong N+\partial V where ≅\cong denotes the relation of being isomorphic in DD. In particular, isomorphic objects are cobordant. Being cobordant is an equivalence relation and for any object MM in DD, one has ∂M∼ cob0\partial M\sim_{cob} 0. Objects of the form ∂M\partial M where MM is an object in DD are said to be boundaries and the objects VV such that ∂V=0\partial V = 0 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 MM is a boundary and M≅NM\cong N then NN is also a boundary. To summarize, 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 Ω(D,∂,i)\Omega(D,\partial,i), which is called the cobordism semigroup (although it is a monoid) of the cobordism category (D,∂,i)(D,\partial,i). The Thom group? 𝒩 *\mathcal{N}_* of cobordism classes of unoriented compact smooth manifolds is an example where D=Diff cD=Diff_c.

Literature

• Robert E. Stong, Notes on cobordism theory, Princeton University Press 1968 (Russian transl., Mir 1973)

Compare also some other nlab entries on cobordism theory: extended cobordism, cobordism, cobordism hypothesis, generalized tangle hypothesis, (infinity,n)-category of cobordisms.