Cobordism. Note Quote.

Objects and arrows in general categories can be very different from sets and functions. The category which quantum field theory concerns itself with is called nCob, the ‘n dimensional cobordism category’, and the rough definition is as follows. Objects are oriented closed (that is, compact and without boundary) (n − 1)-manifolds Σ, and arrows M : Σ → Σ′ are compact oriented n-manifolds M which are cobordisms from Σ to Σ′. Composition of cobordisms M : Σ → Σ′ and N : Σ′ → Σ′′ is defined by gluing M to N along Σ′.

Let M be an oriented n-manifold with boundary ∂M. Then one assigns an induced orientation to the connected components Σ of ∂M by the following procedure. For x ∈ Σ, let (v₁,…,v_n−1,v_n) be a positive basis for T_xM chosen in such a way that (v₁,…,v_n−1) ∈ T_xΣ. It makes sense to ask whether v_n points inward or outward from M. If it points inward, then an orientation for Σ is defined by specifying that (v₁, . . . , v_n−1) is a positive basis for T_xΣ. If M is one dimensional, then x ∈ ∂M is defined to have positive orientation if a positive vector in T_xM points into M, otherwise it is defined to have negative orientation.

Let Σ and Σ′ be closed oriented (n − 1)-manifolds. An cobordism from Σ to Σ′ is a compact oriented n-manifold M together with smooth maps

∑ →ⁱ M ^i’←∑’

where i is a orientation preserving diffeomorphism of Σ onto i(Σ) ⊂ ∂M, i′ is an orientation reversing diffeomorphism of Σ′ onto i′(Σ′) ⊂ ∂M, such that i(Σ) and i′(Σ′) (called the in- and out-boundaries respectively) are disjoint and exhaust ∂M. Observe that the empty set φ can be considered as an (n − 1)-manifold.

One can view M as interpolating from Σ to Σ′. An important property of cobordisms is that they can be glued together. Let M : Σ₀ → Σ1 and M′ : Σ₁ → Σ₂ be cobordisms,

∑₀ →^i₀ M ^i₁ ← ∑₁             ∑₁ →^i’₁ M ^i₂ ← ∑₂

Then we can form a composite cobordism M′ ◦ M : Σ → Σ by gluing M to M′ using i′ ◦ i − 1 : ∂M → ∂M′:

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