ScalingStacks

Consider the surfaces $S_{2}^{1},S_{1}^{2},S_{0}^{1},S_{1}^{0},S_{2}^{2}$ and $S_{1}^{1}$, depicted below

Each of these surfaces $S_{a}^{b}$ defines a cobordism from a union of $a$ circles to a union of $b$ circles. We denote by $\mathcal{M}$ the category whose objects are closed one-dimensional manifolds and morphisms are two-dimensional cobordisms between these manifolds generated by the above cobordisms. Specifically, objects of $\mathcal{M}$ are enumerated by nonnegative integers $\mbox{Ob}(\mathcal{M})=\{\overline{n}|n\in\mathbb{Z}_{+}\}.$ A morphism between $\overline{n}$ and $\overline{m}$ is a compact oriented surface $S$ with boundary being the union of $n+m$ circles. The boundary circles are split into two sets $\partial_{0}S$ and $\partial_{1}S$ with $\partial_{0}S$ containing $n$ and $\partial_{1}S$ containing $m$ circles. An ordering of elements of each of these two sets is fixed. The surface $S$ is presented as a concatenation of disjoint unions of elementary surfaces, depicted above. Morphisms are composed in the usual way by gluing boundary circles. Two morphisms are equal if the surfaces $S,T$ representing these morphisms are diffeomorphic via a diffeomorphism that extends the identification $\partial_{0}S\cong\partial_{0}T,\partial_{1}S\cong\partial_{1}T$ of their boundaries. $\mathcal{M}$ is a monoidal category with tensor product of morphisms defined by taking the disjoint union of surfaces.

Let us construct a monoidal functor from $\mathcal{M}$ to the category $R{\mbox{-mod}_{0}}$ of graded $R$-modules and graded module maps. Assign graded $R$-module $A^{\otimes n}$ to the object $\overline{n}$ and to the elementary surfaces $S_{2}^{1},S_{1}^{2},S_{0}^{1},S_{1}^{0},S_{2}^{2},S_{1}^{1}$ assign morphisms $m,\Delta,\iota,\epsilon,\mbox{Perm},\mbox{Id}$

where $\mbox{Perm}:A\otimes A\to A\otimes A$ is the permutation map, $\mbox{Perm}(u\otimes v)=v\otimes u,$ and Id is the identity map $\mbox{Id}:A\to A.$

To check that $F$ is well-defined one must verify that for any two ways to glue an arbitrary surface $S$ in ${\mathrm{Mor}}(\mathcal{M})$ from copies of these six elementary surfaces, the two maps of $R$-modules, defined by these two decompositions of $S,$ coincide. This follows from the commutative algebra and cocommutative coalgebra axioms of $A$ and the identity (16).

Remark: Suppose that a surface $S\in\mathcal{M}$ contains a punctured genus two surface as a subsurface. Then $F(S)$ is the zero map. Indeed, we only need to check this when $S$ has genus two and one boundary component. Then that $F(S)=0$ follows from $m\circ\Delta\circ m\circ\Delta=0.$

Maps $F(S_{2}^{1}),F(S_{1}^{2}),F(S_{0}^{1}),F(S_{1}^{0}),F(S_{2}^{2})$ and $F(S_{1}^{1})$ between tensor powers of $A$ are graded relative to ${\mathrm{deg}}$ with degrees

From this we deduce