We study the Ginzburg dg algebra Γ_T associated with the quiver with potential arising from a triangulation T of a decorated marked surface S_△⁠, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories D_{fd}(Γ_T)⁠, denoted by D_{fd}(S_△)⁠. As an application, we show that the spherical twist group ST(S_△) associated with D_{fd}(S_△) acts faithfully on its space of stability conditions.

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.

In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring k, via introducing inductive functors which generalize important properties of shift functors of FI-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory are equipped with inductive functors, and hence the finiteness of Castelnuovo-Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when k is a field of characteristic 0.

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this article. Let $\cal A$ be an extension closed subcategory of an extriangulated category $\C$. Then the quotient category $\cal M:=\cal A/\X$ carries naturally a triangulated structure whenever $(\cal A,\cal A)$ forms an $\X$-mutation pair. This result unifies many previous constructions of triangulated quotient categories, and using it gives a classification of thick triangulated subcategories of pretriangulated category $\C/\X$, where $\X$ is functorially finite in $\C$. When $\C$ has Auslander-Reiten translation $\tau$, we prove that for a functorially finite subcategory $\X$ of $\C$ containing projectives and injectives, $\C/\X$ is a triangulated category if and only if $(\C,\C)$ is $\X-$mutation, and if and only if $\tau \underline{\X}=\overline{\X}.$ This generalizes a result by J{\o}rgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory $\X$ of the extriangulated category $\C$, $\C$ admits a new extriangulated structure such that $\C$ is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present examples that extriangulated categories are neither exact categories nor triangulated categories.\

We study the functorially finite maximal rigid subcategories in 2-CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously functorially finite and maximal rigid; we prove that the converse is true if the 2-CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of Keller and Reiten (2007) [KR], we prove that any functorially finite maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via simple mutations, then their endomorphism algebras have the same representation type.