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

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category D(Γ_S) associated to the surface and the corresponding Seidel–Thomas braid group of D(Γ_S) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

We classify the Ext-quivers of hearts in the bounded derived category D(A_n) and the finite-dimensional derived category D(\Gamma_N A_n) of the Calabi-Yau-N Ginzburg algebra D(\Gamma_N A_n). This provides the classification for Buan-Thomas' colored quiver for higher clusters of A-type. We also give explicit combinatorial constructions from a binary tree with n+2 leaves to a torsion pair in mod k\overrightarrow{A_n} and a cluster tilting set in the corresponding cluster category, for the straight oriented A-type quiver \overrightarrow{A_n}. As an application, we show that the orientation of the n-dimensional ssociahedron induced by poset structure of binary trees coincides with the orientation induced by poset structure of torsion pairs in mod k\overrightarrow{A_n} (under the correspondence above).