Integrate on a closed stratified surface

Abstract

We prove that the Drinfeld center centralized by a symmetric fusion category is a symmetric monoidal functor if we choose proper domain and codomain categories. We also compute the factorization homology of stratified surfaces with coefficients satisfying certain anomaly-free conditions.

Figures (3)

• Figure 1: Fig. \ref{['fig:sub1']} depicts a stratified 2-disk $K=(\mathbb{R}^2; \mathbb{R} \cup \mathbb{R})$ with a coefficient system $A_K$ determined by 2-disk algebras $\EuScript{C}, \EuScript{D}, \EuScript{B}$ for 2-cells and 1-disk algebras $\EuScript{M}, \EuScript{N}$ for 1-cells. Fig. \ref{['fig:sub2']} depicts a stratified 2-disk $K'=(\mathbb{R}^2; \mathbb{R})$ with a coefficient system $A_{K'}$ determined by 2-disk algebras $\EuScript{C}, \EuScript{B}$ for 2-cells and a 1-disk algebra $\EuScript{M} \otimes_{\EuScript{D}} \EuScript{N}^{\mathrm{rev}}$ for the 1-cell. $K'$ is obtained by fusing two 1-cells $\EuScript{M}, \EuScript{N}$ and a 2-cell $\EuScript{D}$ to a 1-cell $\EuScript{M} \otimes_{\EuScript{D}} \EuScript{N}^{\mathrm{rev}}$.
• Figure 2: These two figures depict a process of contracting a 1-cell to a 0-cell with proper new target labels such that the value of factorization homology is not changed. Fig. \ref{['fig:sub11']} depicts a stratified 2-disk $M$ with a coefficient system $A_M$ determined by its target labels. Fig. \ref{['fig:sub22']} depicts another stratified 2-disk $M'$ with a coefficient system $A_{M'}$ determined by its target labels. $M'$ is obtained by contracting the internal edge $\EuScript{L}$ in Fig. \ref{['fig:sub11']} to a point.
• Figure 3: These two figures depict a process of merging two 1-cells and one 2-cell to a single 1-cell with proper new target labels such that the value of factorization homology is not changed.

Theorems & Definitions (81)

• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Example 2.6
• Definition 2.7
• Proposition 2.8