The Bar-Natan cobordism category

Let an object be a finite disjoint union of smooth embedded circles in the oriented plane ${\mathbb{R}}^2$, including the empty union. A cobordism from $C_0$ to $C_1$ is a compact smooth orientable surface properly embedded in ${\mathbb{R}}^2\times [0,1]$, with boundary $C_0\times \{0\}\bigsqcup C_1\times \{1\}$, product collars at those boundaries, and no other boundary. Surfaces are identified up to ambient isotopy fixed on the boundary collars. Compose cobordisms by gluing along their common boundary circles; the identity of $C$ is $C\times [0,1]$, and disjoint union gives the monoidal product.

Take the free abelian group on these isotopy classes as each morphism group, and extend composition bilinearly. Let ${\mathsf{C}\mathsf{o}\mathsf{b}}_3$ be the resulting preadditive monoidal category before local relations are imposed. Quotient its morphism groups by the following local relations, which are required to hold inside every surrounding cobordism. A sphere component is zero (the S relation). Adding a disjoint torus component multiplies a cobordism by $2$ (the T relation). For four disjoint disk patches numbered $1,2,3,4$, write $S_{ij}$ for the surface obtained by deleting the interiors of patches $i,j$ and joining their boundary circles by an orientable tube; the four-tube relation is $S_{12}+S_{34}=S_{13}+S_{24}$. The resulting preadditive monoidal quotient category is denoted ${\mathsf{C}\mathsf{o}\mathsf{b}}_{BN}$.

Bar-Natan, Khovanov's homology for tangles and cobordisms, §§3–4 and 7

Kauffman, An Introduction to Khovanov Homology, §5