# Decorated marked surfaces (part B): topological realizations

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

# Decorated marked surfaces (part B): topological realizations

, Volume 288 (2) – Mar 23, 2017
15 pages

/lp/springer_journal/decorated-marked-surfaces-part-b-topological-realizations-0Xy7L9Cu6m
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1876-1
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Mathematische ZeitschriftSpringer Journals

Published: Mar 23, 2017

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