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86
Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapt ..."
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Cited by 75 (9 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
Introduction to Ainfinity algebras and modules
 Homology, Homotopy and Applications
"... Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras, ..."
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Cited by 68 (6 self)
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Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞algebras,
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Clustertilted algebras are Gorenstein and stably
 CalabiYau, Adv. Math
"... Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that ..."
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Cited by 56 (12 self)
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Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that clustertilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3CalabiYau duality over non stable endomorphism rings. This strengthens and generalizes the Extgroup symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative CalabiYau duality from 2CalabiYau to dCalabiYau categories. We show how to produce many examples of dcluster tilted algebras. 1.
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
On the Cyclic Homology of Exact Categories
 JPAA
"... The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective m ..."
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Cited by 45 (1 self)
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The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show that McCarthy's theory cannot be both compatible with localizations and invariant under functors inducing equivalences in the derived category. This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.
RIGID MODULES OVER PREPROJECTIVE ALGEBRAS II: THE Kacmoody Case
, 2007
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials ..."
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Cited by 40 (7 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig’s semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric KacMoody Lie algebra g associated to the quiver Q.
Higher dimensional AuslanderReiten theory on maximal orthogonal subcategories
, 2005
"... We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them. ..."
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Cited by 37 (11 self)
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We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them.
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Cited by 32 (5 self)
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 30 (0 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data