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Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded categories $A, B$, and introduce the orbit bimodule $M/G$ and the smash product bimodule $N\# G$. We will show that these constructions are inverses to each other. This will be applied to Morita equivalence, stable equivalences of Morita type, singular equivalences of Morita type, and singular equivalences of Morita type with level., Comment: 40 pages. Sect. 6 changed and corrected

We define two notions. The first one is a $compression\ system$ $\xi$ for a finite poset $\mathbf{P}$, which assigns each interval subposet $I$ to a poset morphism $\xi_I \colon Q_I \to \mathbf{P}$ satisfying some conditions, where $Q_I$ is a connected finite poset. An example is given by the $total$ compression system that assigns each $I$ to the inclusion of $I$ into $\mathbf{P}$. The second one is an $I$-$rank$ of a persistence module $M$ under $\xi$, the family of which is called the $interval\ rank\ invariant$ of $M$ under $\xi$. A compression system $\xi$ makes it possible to define the $interval\ replacement$ (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant, which is a stronger property than the preservation of the usual rank invariant. Moreover, to know what is preserved explicitly, we will give a formula of the $I$-rank of $M$ under $\xi$ in terms of the structure linear maps of $M$ for any compression system $\xi$, and give a sufficient condition for the $I$-rank of $M$ under $\xi$ to coincide with that under the total compression system, the value of which is equal to the generalized rank invariant introduced by Kim--M\'emoli., Comment: Replaced with a revised version. Major updates include (1) providing a more general formula for computing the interval rank invariant of persistence modules under any compression system $\xi$; (2) revising typos and changing some notions; (3) providing a link to our GitHub program on the interval replacement computation

Throughout this paper $G$ is a fixed group, and $k$ is a fixed field. All categories are assumed to be $k$-linear. First we give a systematic way to induce $G$-precoverings by adjoint functors using a 2-categorical machinery, which unifies many similar constructions of $G$-precoverings. Now let $\mathcal{C}$ be a skeletally small category with a $G$-action, $\mathcal{C}/G$ the orbit category of $\mathcal{C}$, $(P, \phi) : \mathcal{C} \rightarrow \mathcal{C}/G$ the canonical $G$-covering, and $\mathrm{mod}\mbox{-} \mathcal{C}$, $\mathrm{mod}\mbox{-} (\mathcal{C}/G)$ the categories of finitely generated modules over $\mathcal{C}, \mathcal{C}/G$, respectively. Then it is well known that there exists a canonical G-precovering $(P., \phi.) : \mathrm{mod}\mbox{-} \mathcal{C} \rightarrow \mathrm{mod}\mbox{-} (\mathcal{C}/G)$. By applying the machinery above to this $(P., \phi.)$, new $G$-precoverings $(\mathrm{mod}\mbox{-} \mathcal{C}) / S \rightarrow (\mathrm{mod}\mbox{-} \mathcal{C}/G)/S'$ are induced between the factor categories or localizations of $\mathrm{mod}\mbox{-} \mathcal{C}$ and $\mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively. This is further applied to the morphism category $\mathrm{H}(\mathrm{mod}\mbox{-} \mathcal{C})$ of $\mathrm{mod}\mbox{-} \mathcal{C}$ to have a $G$-precovering $\mathrm{fp}(\mathcal{K}) \rightarrow \mathrm{fp}(\mathcal{K}')$ between the categories of finitely presented modules over suitable subcategories $\mathcal{K}$ and $\mathcal{K}'$ of $\mathrm{mod}\mbox{-}\mathcal{C}$ and $ \mathrm{mod}\mbox{-} \mathcal{C}/G$, respectively.

Let $G$ be a finitely generated right $A$-module for a finite-dimensional algebra $A$ over a filed $\Bbbk$, and $\mathcal{I}$ the additive closure of $G$. We will define a $\mathcal{I}$-relative Koszul coresolution $\mathcal{K}^{\bullet}(V)$ of an indecomposable direct summand $V$ of $G$, and show that for a finitely generated $A$-module $M$, the $\mathcal{I}$-relative $i$-th Betti number for $M$ at $V$ is given as the $\Bbbk$-dimension of the $i$-th homology of the $\mathcal{I}$-relative Koszul complex $\mathcal{K}_V(M)_{\bullet}:=\operatorname{Hom}_A(\mathcal{K}^{\bullet}(V),M)$ of $M$ at $V$ for all $i \ge 0$. This is applied to investigate the minimal interval resolution/coresolution of a persistence module $M$, e.g., to check the interval decomposability of $M$, and to compute the interval approximation of $M$., Comment: 36 pages. Main Theorem (3.9) is improved, now without the generator/cogenerator assumption on the module G. In the introduction, an outline of the solution is written in more detail. Some examples were added that treat the case where the endomorphism algebra of G is isomorphic to a lower semi-lattice

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

A diagram consisting of differential graded (dg for short) categories and dg functors is formulated in this paper as a colax functor $X$ from a small category $I$ to the 2-category k-dgCat of small dg categories, dg functors and dg natural transformations for a fixed commutative ring $\mathbb{k}$. If $I$ is a group regarded as a category with only one object $*$, then $X$ is nothing but a colax action of the group $I$ on the dg category $X(*)$. In this sense, this $X$ can be regarded as a generalization of a dg category with a colax action of a group. We define a notion of standard derived equivalence between such colax functors by generalizing the corresponding notion between dg categories with a group action. Our first main result gives some characterizations of this notion, one of which is given in terms of generalized versions of a tilting object and a quasi-equivalence. On the other hand, for such a colax functor $X$, the dg categories $X(i)$ with $i$ objects of $I$ can be glued together to have a single dg category $\int X$, called the Grothendieck construction of $X$. Our second main result insists that for such colax functors $X$ and $X'$, the Grothendieck construction $\int X'$ is derived equivalent to $\int X$ if there exists a standard derived equivalence from $X'$ to $X$. These results generalize the main results of \cite{Asa-a} and \cite{Asa-13} to the dg case, respectively. These are new even for dg categories with group actions. In particular, the second result gives a new tool to show the derived equivalence between the orbit categories of dg categories with group actions, which will be illustrated in some examples., Comment: Fix some errors, new MSC class, new theroem is added, new abstract, rewrite the introduction, all the comments are wellcome. arXiv admin note: text overlap with arXiv:0906.0761 by other authors

In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module $M$, we propose an "interval-decomposable replacement" $\delta^{\ast}(M)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that $M$ is interval-decomposable if and only if $\delta^{\ast}(M)$ is equal to $M$ in the split Grothendieck group. Furthermore, even for modules $M$ not necessarily interval-decomposable, $\delta^{\ast}(M)$ preserves the dimension vector and the rank invariant of $M$. In addition, we provide an algorithm to compute $\delta^{\ast}(M)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\times n$ case)., Comment: 43 pages, changed term "interval-decomposable approximation" to "interval-decomposable replacement"

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $\tau$-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics., Comment: 30 pages

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the ``equioriented'' commutative $2$D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an $n$D persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For $2$D persistence modules, we provide an algorithm together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative $2$D grid. We also propose several heuristics to speed up the computation., Comment: 37 pages

Let $\Bbbk$ be a field, $G$ a group, and $(Q, I)$ a bound quiver. A map $W\colon Q_1 \to G$ is called a $G$-weight on $Q$, which defines a $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$, and $W$ is called homogeneous if $I$ is a homogeneous ideal of the $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$. Then we have a $G$-graded $\Bbbk$-category $\Bbbk(Q, I, W):= \Bbbk(Q, W)/I$. We can then form a smash product $\Bbbk(Q, I, W)\# G$ of $\Bbbk(Q, I, W)$ and $G$, which canonically defines a Galois covering $\Bbbk(Q, I, W)\# G \to \Bbbk(Q, I)$ with group $G$ (we will see that all such Galois coverings to $\Bbbk(Q, I)$ have this form for some $W$). First we give a quiver presentation $\Bbbk(Q_{G, W}, I_{G, W}) \cong \Bbbk(Q, I, W)\# G$ of the smash product $\Bbbk(Q, I, W)\# G$. Next if $(Q, I, W)$ is defined by a Brauer graph with an admissible weight then the smash product $\Bbbk(Q, I, W)\# G$ is again defined by a Brauer graph, which will be computed explicitly. The computation is simplified by introducing a concept of Brauer permutations as an intermediate one between Brauer graphs and Brauer bound quivers. This extends and simplifies the result by Green--Schroll--Snashall on the computation of coverings of Brauer graphs, which dealt with the case that $G$ is a finite abelian group, while in our case $G$ is an arbitrary group. In particular, it enables us to delete all cycles in Brauer graphs to transform it to an infinite Brauer tree., Comment: 29 pages, 29 figures. The admissibility on group actions was missed in the first version. This was added, and miner typos were corrected in the second version. Some miner English improvemnets were made. This is the final version, to appear in Communications in Algebra

The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence module $M$ on $CL_n(\tau)$ as a morphism between zigzag modules, which can be expressed in a block matrix form. For the representation finite case ($n\leq 4)$, we provide an algorithm that uses certain permissible row and column operations to compute a normal form of the block matrix. In this form an indecomposable decomposition of $M$, and thus its persistence diagram, is obtained., Comment: 31 pages. Updated Affiliations. This is a pre-print of an article published in Japan Journal of Industrial and Applied Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s13160-018-0331-y

Let $A$ be a finite-dimensional algebra over an algebraically closed field $\Bbbk$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mr\'oz. As an example we apply this formula to the Kronecker algebra $A$ and give an explicit formula to compute the indecomposable decomposition of $M$, which enables us to make a computer program.

The Grothendieck construction of a diagram $X$ of categories can be seen as a process to construct a single category $\Gr(X)$ by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category $I$ to the 2-category $\kCat$ of small $\k$-categories for a fixed commutative ring $\k$. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors $X, X' \colon I \to \kCat$ are derived equivalent if there is a derived equivalence from $X(i)$ to $X'(i)$ for all objects $i$ in $I$ satisfying some "$I$-equivariance" conditions. In this paper we glue the derived equivalences between $X(i)$ and $X'(i)$ together to obtain a derived equivalence between Grothendieck constructions $\Gr(X)$ and $\Gr(X')$, which shows that if colax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two $\k$-categories with a $G$-action for a group $G$ are "$G$-equivariantly" derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two $\Bbbk$-algebras $A$ and $A'$ are derived equivalent, then so are the path categories $AQ$ and $A'Q$ for any quiver $Q$; so are the incidence categories $AS$ and $A'S$ for any poset $S$; and so are the monoid algebras $AG$ and $A'G$ for any monoid $G$. Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence., Comment: 28 pages. 2nd version: many changes with oplax --> colax. 3rd version: minor changes including "The k-flatness assumption was added to apply Keller's theorem on derived equivalences of categories."

We will give quiver presentations of the Grothendieck constructions of functors from a small category to the 2-category of $\Bbbk$-categories for a commutative ring $\Bbbk$., Comment: 16 pages, 12 figures

Let $\Bbbk$ be a commutative ring and $I$ a category. As a generalization of a $\Bbbk$-category with a (pseudo) action of a group we consider a family of $\Bbbk$-categories with a (pseudo, lax, or oplax) action of $I$, namely an oplax functor from $I$ to the 2-category of small $\Bbbk$-categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors., Comment: 23 pages, ver 2: A construction of oplax functors using a comonad is added by generalizing a tiny example that should be corrected either by putting a relation\beta'\alpha'= \id_{x'} on X(2) or changing the definition of X(a) to "X(a)(z):= y' for z=x,y and X(a)(\gamma):= \id_{y'} for \gamma = \id_x, \id_y, \alpha, \beta" in Section 2. Some references were added

Given a group $G$, we define suitable 2-categorical structures on the class of all small categories with $G$-actions and on the class of all small $G$-graded categories, and prove that 2-categorical extensions of the orbit category construction and of the smash product construction turn out to be 2-equivalences (2-quasi-inverses to each other), which extends the Cohen-Montgomery duality., Comment: 31 pages. I moved the Sec of G-GrCat into Sec 3, and added Lem 5.6. I added more explanations in the proof of Cor 7.6 with (7.5). I added Def 7.7 and Lem 7.8 with the necessary additional assumptions in Props 7.9 and 7.11. I added Lem 8.8 with a short proof, Rmk 8.9 and the proof of Lem 8.10. The final publication is available at Springer via http://dx.doi.org/10.1007/s10485-015-9416-9

Let $G$ be a group acting on a category $\mathcal{C}$. We give a definition for a functor $F\colon \mathcal{C} \to \mathcal{C}'$ to be a $G$-covering and three constructions of the orbit category $\mathcal{C}/G$, which generalizes the notion of a Galois covering of locally finite-dimensional categories with group $G$ whose action on $\mathcal{C}$ is free and locally bonded defined by Gabriel. Here $\mathcal{C}/G$ is defined for any category $\mathcal{C}$ and we do not require that the action of $G$ is free or locally bounded. We show that a $G$-covering is a universal "$G$-invariant" functor and is essentially given by the canonical functor $\mathcal{C} \to \mathcal{C}/G$. By using this we improve a covering technique for derived equivalence. Also we prove theorems describing the relationships between smash product construction and the orbit category construction by Cibils and Marcos (2006) without the assumption that the $G$-action is free. The orbit category construction by a cyclic group generated by an auto-equivalence modulo natural isomorphisms (e.g., the construction of cluster categories) is justified by a notion of the "colimit orbit category". In addition, we give a presentation of the orbit category of a category with a monoid action by a quiver with relations, which enables us to calculate many examples., Comment: Title changed. Definitions of $\mathrm{Mod}^G\mathcal{C}$ and $\mathrm{Mod}_G\mathcal{B}$ in section 6 were corrected. Proof of Theorem 8.1 is slightly changed to make it more readable. Other minor changes

For each simply-laced Dynkin graph $\Delta$ we realize the simple complex Lie algebra of type $\Delta$ as a quotient algebra of the complex degenerate composition Lie algebra $L(A)_{1}^{\mathbb{C}}$ of a domestic canonical algebra $A$ of type $\Delta$ by some ideal $I$ of $L(A)_{1}^{\mathbb{C}}$ that is defined via the Hall algebra of $A$, and give an explicit form of $I$. Moreover, we show that each root space of $L(A)_{1}^{\mathbb{C}}/I$ has a basis given by the coset of an indecomposable $A$-module $M$ with root easily computed by the dimension vector of $M$., Comment: 43 pages, 5 figures, revised version