Combinatorial skeletons of 2-cobordism and annular categories with applications to equational logic

We introduce a complete set of combinatorial data that encode the category $2\mathfrak{C}\mathfrak{o}\mathfrak{b}$ of all $2$-cobordisms. As an application, we show that the local monoids of $2\mathfrak{C}\mathfrak{o}\mathfrak{b}$ do not have finitely axiomatizable equational theories. As yet another application, we construct a von-Neumann-regular extension of this category. Similar results are provided for the topological annular category and various quotients of the latter, like the affine Temperley–Lieb category.