ScalingStacks

Let $\mathcal{M}_{1}$ be the category whose objects are one-dimensional manifolds that are unions of a finite number of circles and one interval. An ordering of the ends of this interval is fixed. Morphisms between objects $\alpha$ and $\beta$ of $\mathcal{M}_{1}$ are oriented surfaces whose boundary is the union of $\alpha,\beta$ and 2 intervals that join corresponding ends of the intervals of $\alpha$ and $\beta.$ An example is depicted below.

This surface represents a morphism from an interval to a union of a circle and an interval. We require that a surface can be presented as a composition of disjoint unions of surfaces $S_{2}^{1},S_{1}^{2},S_{0}^{1},S_{1}^{0},S_{2}^{2},S_{1}^{1},$ defined in Section 2.3, and surfaces $T_{1},T_{2},$ depicted below

We compose morphisms in this category by concatenating surfaces.

Category $\mathcal{M}_{1}$ is a module-category over $\mathcal{M},$ defined in Section 2.3. The bifunctor $\mathcal{M}\otimes\mathcal{M}_{1}\to\mathcal{M}_{1}$ is defined on objects and morphisms by taking disjoint unions.

Recall from the previous section that $A$-mod denotes the category of graded $A$-modules and graded homomorphisms. Given a graded $A$-module $M$, define a monoidal functor

by assigning the graded $A$-module $A^{\otimes n}\otimes M$ to a union of $n$ circles and one interval, maps $\Delta_{M}$ and $m_{M}$ (defined in the previous section) to the elementary surfaces $T_{1}$ and $T_{2}$:

To the other six elementary surfaces $S_{2}^{1},S_{1}^{2},S_{0}^{1},S_{1}^{0},S_{2}^{2},S_{1}^{1}$ (Section 2.3) associate the same maps as for the functor $F$: