## A construction of an A∞-category (2002)

### Cached

### Download Links

Citations: | 6 - 3 self |

### BibTeX

@MISC{Lyubashenko02aconstruction,

author = {Volodymyr Lyubashenko and Sergiy Ovsienko},

title = { A construction of an A∞-category},

year = {2002}

}

### OpenURL

### Abstract

We construct an A∞-category D(C|B) from a given A∞-category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(C|B) to construct an A∞-functor of K-injective resolutions of a complex. The conventional derived category is obtained as the 0-th cohomology of the quotient of differential graded category of complexes over acyclic complexes.

### Citations

391 | Basic concepts of enriched category theory
- Kelly
- 2005
(Show Context)
Citation Context ... closed monoidal symmetric category. The inner hom-object is the usual Hom • k (-, -). There is a notion of a category C enriched in K (K-categories, K-functors, K-natural transformations), see Kelly =-=[Kel82]-=-: for all objects X, Y of C C(X, Y ) is an object of K. There is a similar notion of a 2-category enriched in K, or a K-2-category: it consist of a class of objects Ob C, a class of 1-morphisms C(X, Y... |

355 | Homological algebra of Mirror Symmetry
- Kontsevich
- 1994
(Show Context)
Citation Context ...s. The notion of an A∞-category is a natural generalization of A∞-algebras. It arose in connection with Floer homology in Fukaya’s work [Fuk93, Fuk02] and was related by Kontsevich to mirror symmetry =-=[Kon95]-=-. See Keller [Kel01] for a survey on A∞-algebras and categories. In the present article we show that given two A∞-categories A and B, one can construct a third A∞-category A∞(A, B) whose objects are A... |

333 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ...he 2-category of (unital) A∞-categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article =-=[Sta63, I]-=-. In a sequel to this paper [Sta63, II] Stasheff defines also A∞-algebras and their homotopy-bar constructions. These algebras and their applications to topology were actively studied, for instance, b... |

173 | Lagrangian intersection Floer theory -anomaly and obstruction-, (2000, revised version - Fukaya, Oh, et al. - 2006 |

127 |
Resolutions of unbounded complexes
- Spaltenstein
- 1988
(Show Context)
Citation Context ...ssumption is satisfied, when A has enough injectives and C = C + (A), or when A = R -mod, or when O is a sheaf of rings on a topological space, and A is the category of sheaves of left O-modules, see =-=[Spa88]-=-. Assume now that k is a field. Then for any chain complex of k-modules of the form N = sC(X0, X1)⊗k sC(X1, X2) ⊗k · · · ⊗k sC(Xn−1, Xn), n � 0, Xi ∈ Ob C, for any quasi-isomorphism rX : X → Y , and f... |

115 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...gories). A natural A∞-transformation r : f → g : A → B (natural transformation in terms of [Fuk02]) is an A∞-transformation of degree −1 such that rb + br = 0 (that is, (r)B1 = 0). The ω-globular set =-=[Bat98]-=- Aω of A∞-categories is defined as follows: objects (0-morphisms) are A∞-categories A; 1-morphisms are A∞-functors f : A → B; 2-morphisms are natural A∞-transformations r : f → g : A → B; 3-morphisms ... |

100 |
Enhanced triangulated categories
- Bondal, Kapranov
- 1991
(Show Context)
Citation Context ...ategory B. Originally it has been defined by Drinfeld for differential graded categories [Dri02]. Bondal and Kapranov proposed to produce triangulated categories out of differential graded categories =-=[BK90]-=-. Drinfeld’s construction deals with their quotients, in particular, it produces derived categories. The usefulness of A∞-approach is explained by our construction of an A∞-functor, which assigns to a... |

85 |
Die Grundlehren der mathematischen Wissenschaften
- Lane, Homology
- 1963
(Show Context)
Citation Context ...f graded algebra were given their modern names in H. Cartan’s note [Car48]. See Boardman [Boa66] for operad-like approach to signs as opposed to closed symmetric monoidal category picture of Mac Lane =-=[Mac63]-=- (standard sign commutation rule). Combined together, these sign conventions make the number of signs in this paper tolerable. If u : A → C, a ↦→ au, is a chain map, its cone is the complex Cone(u) = ... |

81 | DG quotients of DG categories
- Drinfeld
(Show Context)
Citation Context ...otation, sign conventions, composition convention, etc. used in the article. The ground commutative ring k is not assumed to be a field. This is suggested by the development of homological algebra in =-=[Dri04]-=-. Working over a ring k instead of a field has strong consequences. For instance, one may not hope for Kadeishvili’s theorem on minimal models [Kad82] to hold for all A∞-algebras over k. In the second... |

80 | Sheaves on manifolds, Grundlehren der mathem. Wissenschaft 292 - Kashiwara, Schapira - 1990 |

74 | A∞-algebras and the cyclic bar complex
- Getzler, Jones
- 1990
(Show Context)
Citation Context ...ead. If C is a Z-graded k-module, then sC denotes the same k-module with the grading (sC) d = C d+1 . The “identity” map C → sC of degree −1 is also denoted s. We follow Getzler–Jones sign convention =-=[GJ90]-=- (Quillen’s rule): (x ⊗ y)(f ⊗ g) = (−) yf xf ⊗ yg = (−1) deg y·deg f xf ⊗ yg. A chain complex is called contractible if its identity endomorphism is homotopic to zero. We use the following standard e... |

69 | Introduction to A-infinity algebras and modules
- Keller
(Show Context)
Citation Context ...A∞-category is a natural generalization of A∞-algebras. It arose in connection with Floer homology in Fukaya’s work [Fuk93, Fuk02] and was related by Kontsevich to mirror symmetry [Kon95]. See Keller =-=[Kel01]-=- for a survey on A∞-algebras and categories. In the present article we show that given two A∞-categories A and B, one can construct a third A∞-category A∞(A, B) whose objects are A∞-functors f : A → B... |

68 | homotopy, A ∞ -category, and Floer homologies - Fukaya, Morse - 1993 |

47 | On the cyclic homology of exact categories
- Keller
- 1999
(Show Context)
Citation Context ... the quotient of the differential graded category of complexes over acyclic complexes. In [Dri02] Drinfeld reviews and develops Keller’s construction of the quotient of differential graded categories =-=[Kel99]-=- and gives a new construction of the quotient. This construction consists of two parts. The first part replaces given pair B ⊂ C of a differential graded category C and its full subcategory B with ano... |

40 |
Lyubashenko V V, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with
- Kerler
- 2001
(Show Context)
Citation Context ...hain maps modulo homotopy, and c is its standard symmetry. There is a notion of a category C enriched in V (V-categories, V-functors, V-natural transformations), see Kelly [Kel82], summarized e.g. in =-=[KL01]-=-: for all objects X, Y of C C(X, Y ) is an object of V. Denote by V-Cat the category, whose objects are V-categories and morphisms are V-functors. Since V is symmetric, the category V-Cat is symmetric... |

36 |
The algebraic structure in the homology of an A∞-algebra (in
- Kadeishvili
- 1982
(Show Context)
Citation Context ...sted by the development of homological algebra in [Dri04]. Working over a ring k instead of a field has strong consequences. For instance, one may not hope for Kadeishvili’s theorem on minimal models =-=[Kad82]-=- to hold for all A∞-algebras over k. In the second section we recall or give definitions of the main objects. A k-quiver is such a graph that the set of arrows (morphisms) between two vertices (object... |

30 | homotopy, A∞-category, and Floer homologies, The - Fukaya, Morse - 1993 |

25 |
Sur les A∞-catégories
- Lefèvre-Hasegawa
- 2002
(Show Context)
Citation Context ...uiver A∞(A, B) has several objects. Thus theory of A∞-algebras leads to the theory of A∞-categories. 5.1 Proposition (See also Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS] and Lefèvre-Hasegawa =-=[LH03]-=-). Let A, B be A∞-categories. Then there exists a unique (1,1)coderivation B : TsA∞(A, B) → TsA∞(A, B) of degree 1, such that B0 = 0 and (r 1 ⊗ · · · ⊗ r n )θb = [(r 1 ⊗ · · · ⊗ r n )B]θ + (−) r1 +···... |

20 | Préfaisceaux, SGA 4: Théorie des Topos et Cohomologie Etale des Schémas, Tome 1. Théorie des Topos - Grothendieck, Verdier |

18 | Floer homology and mirror symmetry
- Fukaya
- 2001
(Show Context)
Citation Context ... B, one can construct a third A∞-category A∞(A, B) whose objects are A∞-functors f : A → B, and morphisms are natural A∞-transformations between such functors. This result was also obtained by Fukaya =-=[Fuk]-=- and by Kontsevich and Soibelman [KS], independently and, apparently, earlier. We describe compositions between such categories of A∞-functors, which would allow us to construct a 2-category of unital... |

18 | Deformation theory - Kontsevich, Soibelman |

18 |
Floer homology and mirror symmetry II, Minimal Surfaces, Geometric Analysis and Symplectic Geometry
- Fukaya
- 1999
(Show Context)
Citation Context ... B, one can construct a third A∞-category A∞(A, B) whose objects are A∞-functors f : A → B, and morphisms are natural A∞-transformations between such functors. This result was also obtained by Fukaya =-=[Fuk02]-=- and by Kontsevich and Soibelman [KS], independently and, apparently, earlier. We describe compositions between such categories of A∞-functors, which would allow us to construct a 2-category of unital... |

14 |
Introduction to A-infinity algebras and modules, Homology Homotopy
- Keller
(Show Context)
Citation Context ...n A∞-category is a natural generalization of A∞-algebras. It arose in connection with Floer homology in Fukaya’s work [Fuk93, Fuk] and was related by Kontsevich to mirror symmetry [Kon95]. See Keller =-=[Kel01]-=- for a survey on A∞-algebras and categories. In the present article we show that given two A∞-categories A and B, one can construct a third A∞-category A∞(A, B) whose objects are A∞-functors f : A → B... |

11 |
A survey of definitions of n-category
- Leinster
(Show Context)
Citation Context ...rn the ω-globular set Aω into a weak non-unital ω-category in the sense of some of the existing definitions of the latter. Plenty of such definitions including [Bat98] are listed in Leinster’s survey =-=[Lei02]-=-. We do not try to proceed in this direction. Instead we truncate the ω-globular set to a 2-globular set (that is, we deal with 0-, 1- and 2-morphisms) and we make a 2-category out of it. 7. 2-categor... |

8 |
A∞-categories and non-commutative geometry, in preparation
- Kontsevich, Soibelman
- 2002
(Show Context)
Citation Context ...y A∞(A, B) whose objects are A∞-functors f : A → B, and morphisms are natural A∞-transformations between such functors. This result was also obtained by Fukaya [Fuk02] and by Kontsevich and Soibelman =-=[KS]-=-, independently and, apparently, earlier. We describe compositions between such categories of A∞-functors, which would allow us to construct a 2-category of unital A∞-categories. The latter notion is ... |

5 |
la cohomologie des espaces où opère un groupe. Notions algébriques préliminaires
- Cartan, Sur
- 1948
(Show Context)
Citation Context ...’s rule: (x ⊗ y)(f ⊗ g) = (−) yf xf ⊗ yg = (−1) deg y·deg f xf ⊗ yg. It takes its origin in Koszul’s note [Kos47]. The main notions of graded algebra were given their modern names in H. Cartan’s note =-=[Car48]-=-. See Boardman [Boa66] for operad-like approach to signs as opposed to closed symmetric monoidal category picture of Mac Lane [Mac63] (standard sign commutation rule). Combined together, these sign co... |

5 | Catégories dérivées et foncteurs dérivés, Algebraic Dmodules - Grivel - 1987 |

5 |
Homology of fiber spaces, Uspekhi Mat. Nauk 35
- Smirnov
- 1980
(Show Context)
Citation Context ...uel to this paper [Sta63, II] Stasheff defines also A∞-algebras and their homotopy-bar constructions. These algebras and their applications to topology were actively studied, for instance, by Smirnov =-=[Smi80]-=- and Kadeishvili [Kad80, Kad82]. We adopt some notations of Getzler and Jones [GJ90], which reduce the number of signs in formulas. The notion of an A∞-category is a natural generalization of A∞-algeb... |

3 | On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35 - Kadeishvili - 1980 |

3 |
les opérateurs de dérivation dans un anneau
- Koszul, Sur
- 1947
(Show Context)
Citation Context ...J90], which include the idea to apply operations to complexes with shifted grading, and Koszul’s rule: (x ⊗ y)(f ⊗ g) = (−) yf xf ⊗ yg = (−1) deg y·deg f xf ⊗ yg. It takes its origin in Koszul’s note =-=[Kos47]-=-. The main notions of graded algebra were given their modern names in H. Cartan’s note [Car48]. See Boardman [Boa66] for operad-like approach to signs as opposed to closed symmetric monoidal category ... |

2 |
The principle of signs
- Boardman
- 1966
(Show Context)
Citation Context ...g) = (−) yf xf ⊗ yg = (−1) degy·deg f xf ⊗ yg. It takes its origin in Koszul’s note [Kos47]. The main notions of graded algebra were given their modern names in H. Cartan’s note [Car48]. See Boardman =-=[Boa66]-=- for operad-like approach to signs as opposed to closed symmetric monoidal category picture of Mac Lane [Mac63] (standard sign commutation rule). Combined together, these sign conventions make the num... |

2 |
SGA 4: Théorie des Topos et Cohomologie Etale des Schémas, Tome 1. Théorie des Topos
- Bourbaki, Univers
(Show Context)
Citation Context ...oncise Definition A.1 and an expanded Definition A.3+A.2. In Appendix B we prove that the cone of a homotopical isomorphism is contractible. 41. Conventions We fix a universe U [GV73, Sections 0,1], =-=[Bou73]-=-. Many classes and sets in this paper will mean U -small sets, even if not explicitly mentioned. k denotes a (U -small) unital associative commutative ring. By abuse of notation it denotes also a chai... |

1 |
The principle of signs, Enseignement Math. (2) 12
- Boardman
- 1966
(Show Context)
Citation Context ...) = (−) yf xf ⊗ yg = (−1) deg y·deg f xf ⊗ yg. It takes its origin in Koszul’s note [Kos47]. The main notions of graded algebra were given their modern names in H. Cartan’s note [Car48]. See Boardman =-=[Boa66]-=- for operad-like approach to signs as opposed to closed symmetric monoidal category picture of Mac Lane [Mac63] (standard sign commutation rule). Combined together, these sign conventions make the num... |