Decorated marked surfaces (part B): topological realizations

Abstract

We study categories associated to a decorated marked surface ${\mathbf {S}}_\bigtriangleup$, which is obtained from an unpunctured marked surface $\mathbf {S}$ by adding a set of decorating points. For any triangulation $\mathbf {T}$ of ${\mathbf {S}}_\bigtriangleup$, let $\Gamma _\mathbf {T}$ be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in ${\mathbf {S}}_\bigtriangleup$ and the reachable rigid indecomposables in the perfect derived category ${\text {per}}\,\Gamma _\mathbf {T}$. This is the dual of the bijection, between simple closed arcs in ${\mathbf {S}}_\bigtriangleup$ and reachable spherical objects in the 3-Calabi-Yau category ${\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$, constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient ${\text {per}}\,\Gamma _\mathbf {T}/{\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$ that defines the generalized cluster categories corresponds to the forgetful map ${\mathbf {S}}_\bigtriangleup \rightarrow \mathbf {S}$ (forgetting the decorating points) in a suitable sense.