## DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II

### BibTeX

@MISC{Merkulov_deformationtheory,

author = {Sergei Merkulov and Bruno Vallette},

title = {DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper is the follow-up of [MV08].

### Citations

937 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...iltered colimits. The argument of Page 16 of [GJ94] proves that the category of prop(erad)s has filtered colimits. Since it has pushouts and filtered colimits, it has finite colimits by Chapter IX of =-=[ML98]-=-. We can also construct coequalizers in this category. Since it is an additive category, it is enough to construct cokernels. Let f : P → Q be a morphism of prop(erad)s. Its cokernel is given by the q... |

414 |
An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38
- Weibel
- 1994
(Show Context)
Citation Context ...h is bounded on Ass (2) ∞ (n) for each n. Therefore it converges to the homology of Ass (2) Proof. Let F−p(Ass (2) ∞ ) be the subspace of Ass (2) ∞ ∞ (n) by the Classical Convergence Theorem 5.5.1 of =-=[Wei94]-=-. The first term E 0 pq is equal to the subspace of Ass (2) ∞ spanned by trees with exactly p internal edges between one vertex labelled by • and the other one labelled by �. And the differential d0 i... |

333 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...of elements of I or J are sequentially small with respect to any map in the category of dg S-bimodules. A.2. Transfer Theorem. In the section, we recall the theorem of transfer, mainly due to Quillen =-=[Qui67]-=- Section II.4 (see also S. E. Crans [Cra95] Theorem 3.3 and M. Hovey [Hov99] Proposition 2.1.19). We will use it to endow the category of dg prop(erad)s with a model category structure. Definition (Re... |

229 |
The cohomology structure of an associative ring
- Gerstenhaber
- 1963
(Show Context)
Citation Context ...dg Lie algebra where the boundary map is equal to the twisted differential D(f) = d(f) + [γ, f]. The first definition of this kind of preLie operation appeared in the seminal paper of M. Gerstenhaber =-=[Ger63]-=- in the case of the cohomology of associative algebras. In the case treated by M. Gerstenhaber, the cooperad C is the Koszul dual cooperad As ¡ of the operad As coding associative algebras and the ope... |

172 | Complexe cotangent et déformations I and - Illusie - 1971 |

145 |
Model categories. Mathematical Surveys and Monographs, 63
- Hovey
- 1999
(Show Context)
Citation Context ...(⊕i≥0D ki } {{ } 1 mi,ni ) ⊠P is acyclic. Proposition 22. The pair of adjoint functors P ⊠− Ω−/I ⊠−P : Prop(erad)/P ⇋ Inf. P-biMod : P ⋉ − } {{ } 1 form a Quillen adjunction. Proof. By Lemma 1.3.4 of =-=[Hov99]-=-, it is enough to prove that the right adjoint P ⋉ − preserves fibrations and acyclic fibrations. Let f : M ↠ M ′ be a fibration (resp. acyclic fibration) between two infinitesimal P-bimodules, that i... |

123 |
homotopy algebra and iterated integrals for double loop spaces
- Getzler, Jones, et al.
(Show Context)
Citation Context ...nduces a triple F : S-biMod → S-biMod such that an algebra over it is a prop(erad). Since the underlying category of S-bimodules has limits, the category of prop(erad)s has all limits (Section 1.5 of =-=[GJ94]-=-). □ To prove that the category of prop(erad)s has finite colimits, we first make explicit coproducts and pushouts. This section is the generalization of Section 1.5 of [GJ94] from operads to prop(era... |

116 | Operads and motives in deformation quantization
- Kontsevich
- 1999
(Show Context)
Citation Context ...•(E c , C) is canonically a dg Lie algebra. For finite-dimensional Q and C Theorems 5 and 6 are, of course, equivalent to each other. A morphism of L∞-algebras, (g1, Q1) → (g2, Q2), is, by definition =-=[Kon99]-=-, a morphism, λ : (⊙•(sg1), Q1) → (⊙•(sg2), Q2), of the associated dg coalgebras. It is called a quasi-isomorphism if the composition, sg1 i −→ ⊙ • (sg1) λ −→ ⊙ • (sg2) p −→ sg2 2Note that for any dif... |

116 |
Operads in algebra, topology and physics
- Markl, Shnider, et al.
- 2002
(Show Context)
Citation Context ...avoine in [Bal97] that the deformation complex of algebras over any Koszul operad admits a Lie structure. This statement was made more precise by Markl, Shnider and Stasheff in Section 3.9 Part II of =-=[MSS02]-=- where they proves that this Lie bracket comes from a Prelie product. This result on the level of operads was proved using the space of coderivations of the cofree P ¡ -coalgebra, which is shown to be... |

113 | Deformations of algebras over operads and the Deligne conjecture - Kontsevich, Soibelman |

85 |
Noncommutative differential geometry. Inst. Hautes tudes
- Connes
- 1985
(Show Context)
Citation Context ...ient thanks to the Leibniz } {{ } 1 relation verified by d. □ The module of Kähler differentials of an associative algebra is the non-commutative analog of classical differential forms (see A. Connes =-=[Con85]-=-). Since operads and prop(erad)s can also be used to encode geometry (see [Mer05, Mer06]), the module of Kähler differentials for prop(erad)s seems a promising tool to study non-linear properties in n... |

81 | Homological algebra of homotopy algebras
- Hinich
- 1997
(Show Context)
Citation Context ...he reader to Section 2.5 of [BM03] for the application of this Theorem with stronger and sometimes more convenient hypotheses. Remark that Transfer Theorem 32 was used (and rephrased) by V. Hinich in =-=[Hin97]-=- to provide a model category structure to the category of operads over unbounded chain complexes (see Theorem 2.2.1 of [Hin97] and the corrected version of Theorem 6.6.1 in [Hin03]). M. Spitzweck also... |

74 | A solution of Deligne’s Hochschild cohomology conjecture. Recent progress in homotopy theory
- McClure, Smith, et al.
- 2002
(Show Context)
Citation Context ... controlling deformations of a particular associative algebra □22 SERGEI MERKULOV, BRUNO VALLETTE structure, γ : Ass → EndX, on a vector space X. In the work of McClure-Smith on Deligne’s conjecture =-=[MS02]-=-, an operad Q with a morphism of operads Ass → Q is called a multiplicative operad. The simplicial complex that they define on such an operad is exactly the deformation complex of this map. For the op... |

70 | Homologie cyclique et K-théorie, Astérisque - Karoubi - 1987 |

60 | Koszul duality of operads and homology of partition posets, from: “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic
- Fresse
(Show Context)
Citation Context ...heorem 32. □ A.5. Cofibrations and Cofibrant objects. In this section, we make explicit the cofibrations and the cofibrant objects in the model category of dg prop(erad)s. We refer to the Appendix of =-=[Fre04]-=- for the case of operads. Proposition 37. A map f : P �� �� Q is a cofibration in the model category of dg prop(erad)s if and only if it is a retract of a map P → P ∨ F(S), with isomorphisms on domain... |

58 |
Deformation of Lie algebra structures
- Nijenhuis, Richardson
- 1967
(Show Context)
Citation Context ...is the intrinsic Lie bracket of Stasheff [Sta93]. It is equal to the Lie bracket of Gerstenhaber [Ger63] on Hochschild cochain complex of associative algebras, the Lie bracket of Nijenhuis-Richardson =-=[NR67]-=- on Chevalley-Eilenberg cochain complex of Lie algebras and the Lie bracket of Stasheff on Harrison cochain complex of commutative algebras. It is proven by Balavoine in [Bal97] that the deformation c... |

51 | Combinatorial operad actions on cochains - Berger, Fresse |

50 | Axiomatic homotopy theory for operads
- Berger, Moerdijk
(Show Context)
Citation Context ...a model category structure cofibrantly generated by F (I) as the the set of generating cofibrations and F (J) as the set of generating acyclic cofibrations. We also refer the reader to Section 2.5 of =-=[BM03]-=- for the application of this Theorem with stronger and sometimes more convenient hypotheses. Remark that Transfer Theorem 32 was used (and rephrased) by V. Hinich in [Hin97] to provide a model categor... |

45 |
Infinitesimal computations in topology, Inst. Hautes
- Sullivan
- 1977
(Show Context)
Citation Context ...ds and the later to C∞). We can now choose to work with such cofibrant models. The extra filtration on the space of generators, which appears conceptually here, is similar to the one used by Sullivan =-=[Sul77]-=- in rational homotopy theory and by Markl in [Mar96b] for operads. Let P be a dg properad. Its space of indecomposable elements is the cokernel of the composite map with non-trivial elements, µ : ¯ P ... |

41 |
The intrinsic bracket on the deformation complex of an associative algebra
- STASHEFF
(Show Context)
Citation Context ...erad C is the Koszul dual cooperad As ¡ of the operad As coding associative algebras and the operad P is the endomorphism operad EndA. The induced Lie bracket is the intrinsic Lie bracket of Stasheff =-=[Sta93]-=-. It is equal to the Lie bracket of Gerstenhaber [Ger63] on Hochschild cochain complex of associative algebras, the Lie bracket of Nijenhuis-Richardson [NR67] on Chevalley-Eilenberg cochain complex of... |

37 |
Models for operads
- Markl
- 1996
(Show Context)
Citation Context ...ith such cofibrant models. The extra filtration on the space of generators, which appears conceptually here, is similar to the one used by Sullivan [Sul77] in rational homotopy theory and by Markl in =-=[Mar96b]-=- for operads. Let P be a dg properad. Its space of indecomposable elements is the cokernel of the composite map with non-trivial elements, µ : ¯ P ⊠ ¯ P → ¯ P. The space of indecomposable elements inh... |

34 |
Bialgebra cohomology, deformations, quantum groups and algebraic deformations
- Gerstenhaber, Shack
- 1990
(Show Context)
Citation Context ... case, the complex above is the deformation complex of associative bialgebra, or more generally of AssBi∞-gebra, structure on X. When X is an associative bialgebra, Gerstenhaber and Schack defined in =-=[GS90]-=- a bicomplex whose homology has nice properties with respect to the deformations of the associative bialgebra structure (see also [LM91]). Let us first extend this definition to any properad Q and not... |

30 |
Homologie des Algèbres Commutatives
- André
- 1974
(Show Context)
Citation Context ...d the Appendix □ □DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II 21 when the characteristic of the ground ring is not 0. In this case, one has to use simplicial resolutions like in M. André =-=[And74]-=- and D. Quillen [Qui70]. 3. Examples of deformation theories. In this section, we show that the conceptual deformation theory defined here coincide to well known theories in the case of associative al... |

26 |
On the (co-)homology of commutative rings, Applications of Categorical Algebra
- Quillen
- 1964
(Show Context)
Citation Context ....1. Basic Definition. Let (P, dP) ϕ −→ (Q, dQ) be a morphism of dg prop(erad)s. We would like to define a chain complex with which we could study the deformation theory of this map. Following Quillen =-=[Qui70]-=-, the conceptual method is to take the total right derived functor of the functor Der of derivations from the category of prop(erad)s above Q (see also [Mar96a, VdL02]). That is, we consider a cofibra... |

24 | Modules and Morita theorem for operads - Kapranov, Manin |

22 | Cotangent cohomology of a category and deformations - Markl - 1996 |

22 | algebras and modules in general model categories arXiv: math/0101102v1 Institut de Géométrie, Algèbre et Topologie, EPFL CH-1015
- Spitzweck, Operads
(Show Context)
Citation Context ... corrected version of Theorem 6.6.1 in [Hin03]). M. Spitzweck also applied this theorem to prove a general result about model category structures on categories of algebras over a triple (Theorem 1 of =-=[Spi01]-=-). A.3. Limits and Colimits of prop(erad)s. In this section, we prove that the category of prop(erad)s has all limits and finite colimits. We also make explicit the coproducts and pushouts of prop(era... |

20 | Homology of generalized partition posets
- Vallette
- 2007
(Show Context)
Citation Context ...sm of S-modules Ass (2) ∼ = Ass ∨ Ass. Remark. The example of Ass (2) is also interesting from the viewpoint of Koszul operad. It comes from a set theoretic operad. It is Koszul whereas the method of =-=[Val06a]-=- cannot be applied because Ass (2) is not basic set, that is the composition of operations is not injective. The product � has an “absorbing” effect. In the same way, we define the operad Lie (2) by F... |

19 | Homotopy Gerstenhaber algebras. In Conférence Moshé Flato 1999, Vol - Voronov - 2000 |

17 | A resolution (minimal model) of the PROP for bialgebras
- Markl
(Show Context)
Citation Context ...{} r2 ��� ��� •� � � ��� �� ... }{{} rk where s = (k − 1)(r1 − 1) + (k − 2)(r2 − 1) + . . . + 1(rk − 1), ∆ SU is the Saneblidze-Umble diagonal, and the horizontal line means fraction composition from =-=[Mar06]-=-. The meaning of this part of the differential is clear: it describes A∞-morphisms between an A∞-structure on X and the associated Saneblidze-Umble diagonal A∞-structure on X ⊗ X. Explicitly, this for... |

17 |
On the other side of the bialgebra of chord diagrams
- Tourtchine
(Show Context)
Citation Context ...erad. The simplicial complex that they define on such an operad is exactly the deformation complex of this map. For the operad Q = P oisson, this complex is related to the homology of long knots (see =-=[Tou04]-=-). More generally, Maxim Kontsevich proposed the conjecture that the deformation complex of Ass → EndX is a d + 1-algebra when X is a d-algebra in [Kon99]. This conjecture was proved by Tamarkin in [T... |

16 | PROPped up graph cohomology - Markl, Voronov |

15 |
Deformations of algebras over a quadratic operad
- Balavoine
- 1997
(Show Context)
Citation Context ...of Nijenhuis-Richardson [NR67] on Chevalley-Eilenberg cochain complex of Lie algebras and the Lie bracket of Stasheff on Harrison cochain complex of commutative algebras. It is proven by Balavoine in =-=[Bal97]-=- that the deformation complex of algebras over any Koszul operad admits a Lie structure. This statement was made more precise by Markl, Shnider and Stasheff in Section 3.9 Part II of [MSS02] where the... |

15 | Quillen closed model structures for sheaves
- Crans
- 1995
(Show Context)
Citation Context ...l with respect to any map in the category of dg S-bimodules. A.2. Transfer Theorem. In the section, we recall the theorem of transfer, mainly due to Quillen [Qui67] Section II.4 (see also S. E. Crans =-=[Cra95]-=- Theorem 3.3 and M. Hovey [Hov99] Proposition 2.1.19). We will use it to endow the category of dg prop(erad)s with a model category structure. Definition (Relative I-cell complexes). For every class I... |

15 | On Kontsevich’s Hochschild cohomology conjecture. Preprint, available at http://www.math.umn.edu/∼voronov/igor.ps
- Hu, Kriz, et al.
(Show Context)
Citation Context ...roposed the conjecture that the deformation complex of Ass → EndX is a d + 1-algebra when X is a d-algebra in [Kon99]. This conjecture was proved by Tamarkin in [Tam00], see also Hu, Kriz and Voronov =-=[HKV06]-=-. In this context, this chain complex is often called the Hochschild complex of Q. Since this (co)chain complex comes from the general theory of (co)homology of Quillen, it would be better to call its... |

15 | An L∞-algebra of an unobstructed deformation functor
- Merkulov
(Show Context)
Citation Context ...he homological vector field Q vanishes. From now on we do not distinguish between g and its completion ˆg. To every MC element γ in a filtered L∞-algebra (g, Q) there corresponds, by Theorem 2.6.1 in =-=[Mer00]-=-, a twisted L∞-algebra, (g, Qγ), with Qγ(α) := Q( ∑ n≥0 1 n! γ⊙n ⊙ α) for an arbitrary α ∈ ⊙ ≥1 sg. The geometric meaning of this twisted L∞-structure is simple [Mer00]: if a homological vector field ... |

14 |
products, Koszul duality, Loday algebras and Deligne conjecture
- Vallette, Manin
(Show Context)
Citation Context ...d to be a quotient of C such that the kernel of the associated projection C ↠ I is a subcoalgebra of C. For a complete study of this notion, we refer the reader to Appendix B “Categorical Algebra” of =-=[Val06b]-=-. This notion should not be confused with the notion of coideal used in Hopf algebra theory. Since a Hopf is an algebra and a coalgebra at the same time, a coideal in that sense is a submodule such th... |

12 | Wheeled props, graph complexes and the master equation, arXiv:math/0610683v3
- Markl, Merkulov, et al.
- 2007
(Show Context)
Citation Context ...mplexes, the complex (E0, ∂0) is isomorphic to the tensor product of two isomorphic operadic complexes (one with “time” flow reversed upside down relative to another) which were studied on page 40 of =-=[MMS06]-=- and which have the differential (in notations of that paper) given by d1 • ������ ���� . . . � � n−2 ∑ �� �� �� = (−1) �� i=0 i+1 • ������ ���� .. .. � � �� � �� ��� ��� •� � 1 2 n � �� � 1 i n i+1 i... |

11 | Tamarkin, Another proof of M. Kontsevich formality theorem, math.QA/ 9803025; Formality of chain operad of small squares - E |

10 |
der Laan, Operads up to homotopy and deformations of operad maps
- Van
- 2002
(Show Context)
Citation Context ...field. (iii) As ∂ (n) P = 0 for n > 2 we conclude using formula (5) that Q(n) = 0 for all n > 2. □ A special case of the above Theorem when both P and E are operads was proven earlier by van der Laan =-=[VdL02]-=- using different ideas. The main point of our proof of Theorem 5 is an observation that, for a free prop(erad) P = F(s−1C), the set, MorZ(P, E), of extended morphisms from P to an arbitrary prop(erad)... |

9 | Deformation theory of representations of prop(erad)s
- Merkulov, Vallette
(Show Context)
Citation Context ...er is the follow-up of =-=[MV08]-=-. Introduction In Section 1, we give another, more geometric, interpretation of the L∞-algebra structure on the homotopy convolution properad introduced in Section 4.5 of [MV08]. In Section 2, we define the deformation theory of morphisms of properads following Quillen’s method. It is identified with a homotopy convolution properad, so it carries an natural L∞-algebra struct... |

8 | Generalized operads and their inner cohomomorphisms
- Borisov, Manin
- 2008
(Show Context)
Citation Context ...mits and finite colimits. We also make explicit the coproducts and pushouts of prop(erad)s. Proposition 33. The category of prop(erad)s has all limits. Proof. We recall from D. Borisov and Y.I. Manin =-=[BM06]-=- that the free prop(erad) functor induces a triple F : S-biMod → S-biMod such that an algebra over it is a prop(erad). Since the underlying category of S-bimodules has limits, the category of prop(era... |

5 |
Modular operads and BV
- Barannikov
- 2006
(Show Context)
Citation Context ...its a codifferential such that the right vertical arrow is a morphism of dg coalgebras), then the data (OI, Q) is naturally a differential graded coalgebra which we often call a dg affine scheme (cf. =-=[Bar06]-=-). The coideal may not, in general, be homogeneous so the “weight” gradation, ⊕ n ⊙nX, may not survive in OI. A generic dg affine scheme by no means corresponds to a L∞-algebra but, as we shall see be... |

5 |
crochets de Schouten et cohomologies d’algèbres de
- Kosmann-Schwarzbach, crochet
- 1991
(Show Context)
Citation Context ...The properad of Lie bialgebras is Koszul. Therefore, on the deformation (bi)complex of Lie bialgebras, there is a Lie bracket. The construction of this Lie bracket was given by Kosmann-Schwarzbach in =-=[KS91]-=-. (See also Ciccoli-Guerra [CG03] for the interpretation of this bicomplex in terms of deformations.) 2.4. Definition à la Quillen. In the previous sections, we defined the deformation complex of repr... |

5 |
profile of poisson geometry
- Prop
(Show Context)
Citation Context ... � � • ��� � ����� ���� . . . � � 〉 �� �� � , �� 1 2 n−1 n where signm stands for the sign representation of Sm and 1n for the trivial representation of Sn. The differential is given on generators by =-=[Mer06]-=- ������ ���� . . . � ���� � δ •� ��� ������ ���� . . . � � �� �� ��� � 1 2 m−1 m 1 2 n−1 n = ∑ I 1 ⊔I 2 =(1,...,m) J 1 ⊔J 2 =(1,...,n) |I 1 |≥0,|I 2 |≥1 |J 1 |≥1,|J 2 |≥0 where σ(I1 ⊔ I2) is the sign ... |

3 | A universal enveloping for L∞-algebras
- Baranovsky
- 2007
(Show Context)
Citation Context ...ν), on ⊙ • g obtained via Kontsevich’s “no-wheels” quantization of the associated linear polyvector field ν is called the universal enveloping algebra of the L∞-algebra. In a recent interesting paper =-=[Bar07]-=- Baranovsky also defined a universal enveloping for a L∞algebra g as a certain A∞-structure on the space ⊙•g. In his approach the A∞-structure is constructed with the help of the homological perturbat... |

3 |
The variety of Lie bialgebras
- Ciccoli, Guerra
(Show Context)
Citation Context ... Koszul. Therefore, on the deformation (bi)complex of Lie bialgebras, there is a Lie bracket. The construction of this Lie bracket was given by Kosmann-Schwarzbach in [KS91]. (See also Ciccoli-Guerra =-=[CG03]-=- for the interpretation of this bicomplex in terms of deformations.) 2.4. Definition à la Quillen. In the previous sections, we defined the deformation complex of representations of a prop(erad) P tha... |

3 |
On the Cambell-Baker-Hausdorf deformation quatization of a linear Poisson structure
- Shoikhet
- 2003
(Show Context)
Citation Context ...ν is a linear polyvector field on g∗ satisfying the equation [ν, ν]S = 0, then one can set to zero all contributions to the formality morphism F coming from graphs with closed directed paths (wheels) =-=[Sho03]-=- and the resulting element Fno−wheels(ν) ∈ ⊕n≥0Hompoly(O ⊗n g∗ , Og∗)[[�]] is still Maurer-Cartan. It is easy to check that Fno−wheels(ν) has no summand with weight n = 0 and hence defines an A∞-struc... |

2 |
infinity and contractible dg manifolds
- Nijenhuis
(Show Context)
Citation Context ...sson structures, and LieBi-homology is precisely Poisson homology. In a similar way one can check that our construction of L∞-algebras applied to the minimal resolution of so called pre-Lie2-algebras =-=[Mer05]-=- gives rise to another classical geometric object — the Frölicher-Nijenhuis Lie brackets on the sheaf, TX ⊗ Ω• X , of tangent vector bundle valued differential forms. Thus the associated deformation t... |