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.