## OPERADS AND PROPS (2006)

Citations: | 8 - 0 self |

### BibTeX

@MISC{Markl06operadsand,

author = {Martin Markl},

title = { OPERADS AND PROPS},

year = {2006}

}

### OpenURL

### Abstract

We review definitions and basic properties of operads, PROPs and algebras over these structures.

### Citations

568 |
Deformation Quantization of Poisson Manifolds
- Kontsevich
(Show Context)
Citation Context ...he affirmative answer to the Deligne conjecture [17, 56]. Mathematical physics. The formality mentioned in the previous item implies the existence of the deformation quantization of Poisson manifolds =-=[54]-=-. We must not forget to mention the operadic interpretation of vertex operator algebras [46], string theory [49] and Connes-Kreimer’s approach to renormalization [15]. Operads and multicategories are ... |

363 | Gromov-Witten classes, quantum cohomology and enumerative geometry
- Kontsevich, Manin
- 1994
(Show Context)
Citation Context ...omplex geometry. Applications involve moduli spaces of stable complex algebraic curves of genus zero [34], enumerative geometry, Frobenius manifolds, quantum cohomology and cohomological field theory =-=[55, 71]-=-. The moduli space of genus zero curves exhibits an additional symmetry that leads to a generalization called cyclic operads [32]. Modular operads [33] then describe the combinatorial structure of the... |

346 |
The irreducibility of the space of curves of given genus
- Deligne, Mumford
- 1969
(Show Context)
Citation Context ...als the arithmetic genus of the curve C, thus Mg,n+1 is the coarse moduli space of stable curves of arithmetic genus g with n + 1 marked points. By a result of P. Deligne, F.F. Knudsen and D. Mumford =-=[18, 51, 50]-=-, Mg,n+1 is a projective variety. x1 a1 a2 • • a5 a3 • x2 a4 • x0sOPERADS AND PROPS 35 Observe that, for a curve C ∈ M0,n+1, the graph Γ(C) must necessarily be a tree and all components of C must be s... |

228 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
(Show Context)
Citation Context ...fferential geometry [92, 93]. By the renaissance of operads we mean the first half of the nineties of the last century when several papers which stimulated the rebirth of interest in operads appeared =-=[31, 34, 41, 45, 47, 49, 72]-=-. Let us mention the most important new ideas that emerged during this period. Date: January 25, 2006. The author was supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Cze... |

227 |
The cohomology structure of an associative ring
- Gerstenhaber
- 1963
(Show Context)
Citation Context ...as an abstraction ofs4 M. MARKL type (I) compositions, there exist an alternative approach based on type (II) compositions. This second point of view was formalized in the 1963 papers by Gerstenhaber =-=[29]-=- and Stasheff [106]. A definition that included the symmetric group action was formulated much later in the author’s paper [75] in which the two approaches were also compared. In the presence of opera... |

225 | Renormalization in quantum field theory and the RiemannHilbert problem. II. The β-function, diffeomorphisms and the renormalization group
- Connes, Kreimer
(Show Context)
Citation Context ...quantization of Poisson manifolds [54]. We must not forget to mention the operadic interpretation of vertex operator algebras [46], string theory [49] and Connes-Kreimer’s approach to renormalization =-=[15]-=-. Operads and multicategories are important also for BeilinsonDrinfeld’s theory of chiral algebras [6]. Algebra. Operadic cohomology [1, 26, 31, 34, 83] provides a uniform treatment of all ‘classical’... |

222 |
Rational homotopy theory
- Quillen
- 1969
(Show Context)
Citation Context ...algebras over a reasonable (possibly colored) operad form a model category that generalizes the classical model structures of the categories of dg commutative associative algebras and dg Lie algebras =-=[95, 107]-=-. Operads, in asOPERADS AND PROPS 3 reasonable monoidal model category, themselves form a model category [7, 31] such that algebras over cofibrant operads are homotopy invariant, see also [104]. Minim... |

215 |
The projectivity of the moduli space of stable curves
- Knudsen, Mumford
(Show Context)
Citation Context ...ith ad − bc �= 0. CP 1 ∋ [ξ1, ξ2] ↦→ [aξ1 + bξ2, cξ1 + dξ2] ∈ CP 1 , The moduli space M0,n+1 has, for n ≥ 2, a canonical compactification M0(n) ⊃ M0,n+1 introduced by A. Grothendieck and F.F. Knudsen =-=[16, 50]-=-. The space M0(n) is the moduli space of stable (n + 1)-pointed curves of genus 0: Definition 15. A stable (n + 1)-pointed curve of genus 0 is an object (C; x0, . . ., xn), where C is a (possibly redu... |

198 | Differential analysis on complex manifolds, Graduate Texts - Wells - 1980 |

190 |
Homotopy invariant algebraic structures on topological. spaces
- Boardman, Vogt
(Show Context)
Citation Context ...d to wheeled PROPs introduced in [93]. Deformation theory and homotopy invariant structures in algebra. A concept of homotopy invariant structures in algebra parallel to the classical one in topology =-=[9, 10]-=- was developed in [80]. It was explained in [56, 73, 79] how cofibrant resolutions of operads or PROPs determine a cohomology theory governing deformations of related algebras. In [81], deformations w... |

190 |
Frobenius manifolds, quantum cohomology, and moduli spaces
- Manin
- 1999
(Show Context)
Citation Context ...omplex geometry. Applications involve moduli spaces of stable complex algebraic curves of genus zero [34], enumerative geometry, Frobenius manifolds, quantum cohomology and cohomological field theory =-=[55, 71]-=-. The moduli space of genus zero curves exhibits an additional symmetry that leads to a generalization called cyclic operads [32]. Modular operads [33] then describe the combinatorial structure of the... |

140 |
Formal Noncommutative Symplectic Geometry
- Kontsevich
- 1993
(Show Context)
Citation Context ...sentation of PROP-like structures in enriched monoidal categories can be found in [91]. Graph Complexes. Each cyclic operad P determines a graph complex [33, 77]. As observed earlier by M. Kontsevich =-=[52]-=-, these graph complexes are, for some specific choices of P, closely related to some very interesting objects such as moduli spaces of Riemann surfaces, automorphisms of free groups or primitives in t... |

112 | Deformations of algebras over operads and the Deligne conjecture. Conférence Moshé Flato
- Kontsevich, Soibelman
- 1999
(Show Context)
Citation Context ...rmality of Hochschild cochains of the algebra of functions on smooth manifolds [110] explained in [40] uses obstruction theory for operad algebras and the affirmative answer to the Deligne conjecture =-=[17, 56]-=-. Mathematical physics. The formality mentioned in the previous item implies the existence of the deformation quantization of Poisson manifolds [54]. We must not forget to mention the operadic interpr... |

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...ewed as special kinds of algebraic theory (as can multicategories, if one allows many-sorted theories), see [85]. There are also ‘categorical’ generalizations of operads, e.g. the globular operads of =-=[2]-=- and T-categories of [11]. An interesting presentation of PROP-like structures in enriched monoidal categories can be found in [91]. Graph Complexes. Each cyclic operad P determines a graph complex [3... |

79 | Homological algebra of homotopy algebras
- Hinich
- 1997
(Show Context)
Citation Context ...elation between Koszulness of operads and properties of posets was studied in [27]. Also the concept of the operadic distributive law turned out to be useful [26, 74]. Model structures. It turned out =-=[8, 31, 39, 100]-=- that algebras over a reasonable (possibly colored) operad form a model category that generalizes the classical model structures of the categories of dg commutative associative algebras and dg Lie alg... |

77 |
Two-dimensional conformal geometry and vertex operator algebras
- Huang
- 1997
(Show Context)
Citation Context ...y mentioned in the previous item implies the existence of the deformation quantization of Poisson manifolds [54]. We must not forget to mention the operadic interpretation of vertex operator algebras =-=[46]-=-, string theory [49] and Connes-Kreimer’s approach to renormalization [15]. Operads and multicategories are important also for BeilinsonDrinfeld’s theory of chiral algebras [6]. Algebra. Operadic coho... |

70 | Modular operads
- Getzler, Kapranov
- 1998
(Show Context)
Citation Context ... : � M(g, 0) × � M(h, 0) → � M(g + h, −1), g, h ≥ 0. Modular operads are abstractions of the above structure satisfying a certain additional stability condition. The following definitions, taken from =-=[33]-=-, are made for the category of k-modules, but they can be easily generalized to an arbitrary symmetric monoidal category with finite colimits, whose monoidal product ⊙ is distributive over colimits. L... |

66 |
Adjoint functors and triples
- Eilenberg, Moore
- 1965
(Show Context)
Citation Context ...efines the unit υ of the triple and the counit of the adjunction FG → id induces a natural transformation GFGF → GF which defines the multiplication µ. In fact, it is a theorem of Eilenberg and Moore =-=[20]-=- that all triples arise in this way from adjoint pairs. This is exactly the situation with the free operad and free non-unital operad functors that were described in Section 4. We will show how operad... |

61 | String topology
- Chas, Sullivan
(Show Context)
Citation Context ...s [76]. This fact is crucial for the theory of configuration spaces with summable labels [96]. The cacti operad [117] lies behind the Chas-Sullivan product on the free loop space of a smooth manifold =-=[13]-=-, see also [14]. Tamarkin’s proof of the formality of Hochschild cochains of the algebra of functions on smooth manifolds [110] explained in [40] uses obstruction theory for operad algebras and the af... |

61 | Another proof of M. Kontsevich formality theorem
- Tamarkin
(Show Context)
Citation Context ...ind the Chas-Sullivan product on the free loop space of a smooth manifold [13], see also [14]. Tamarkin’s proof of the formality of Hochschild cochains of the algebra of functions on smooth manifolds =-=[110]-=- explained in [40] uses obstruction theory for operad algebras and the affirmative answer to the Deligne conjecture [17, 56]. Mathematical physics. The formality mentioned in the previous item implies... |

60 | Koszul duality of operads and homology of partition posets, In: “Homotopy theory: relations with algebraic geometry
- Fresse
(Show Context)
Citation Context ...rphisms and homotopy invariance [80]. Operads serve as a natural language for various types of ‘multialgebras’ [64, 65]. Relation between Koszulness of operads and properties of posets was studied in =-=[27]-=-. Also the concept of the operadic distributive law turned out to be useful [26, 74]. Model structures. It turned out [8, 31, 39, 100] that algebras over a reasonable (possibly colored) operad form a ... |

60 |
Coalgebras and bialgebras in combinatorics
- Joni, Rota
(Show Context)
Citation Context ...ense of [108], with one object. Lie bialgebras reviewed in Example 63 are algebras over a dioperad. Another important class of dioperad algebras is recalled in: Example 65. An infinitesimal bialgebra =-=[48]-=- (called in [26, Example 11.7] a mock bialgebra) is a vector space V with an associative multiplication · : V ⊗ V → V and a coassociative comultiplication ∆ : V → V ⊗ V such that ∆(a · b) = �� � a(1) ... |

57 |
Natural associativity and commutativity
- Lane
- 1963
(Show Context)
Citation Context ...ook [86], but a year or more earlier, M. Boardman and R. Vogt [9] described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane =-=[67]-=-. As pointed out in [62], also Lambek’s definition of multicategory [60] (late 1960s) was almost equivalent to what is called today a colored or many-sorted operad. Another important precursor was the... |

55 |
Cyclic homology and the Lie algebra homology of matrices
- Loday, Quillen
- 1984
(Show Context)
Citation Context ...satisfying B([a, b], c) = B(a, [b, c]), for a, b, c ∈ V. For algebras over cyclic operads, one may introduce cyclic cohomology that generalizes the classical cyclic cohomology of associative algebras =-=[12, 66, 109]-=- as the non-abelian derived functor of the universal bilinear form [32], [83, Proposition II.5.26]. Let us close this section by mentioning two examples of operads with other types of higher symmetrie... |

50 | Axiomatic homotopy theory for operads
- Berger, Moerdijk
(Show Context)
Citation Context ...ructures of the categories of dg commutative associative algebras and dg Lie algebras [95, 107]. Operads, in asOPERADS AND PROPS 3 reasonable monoidal model category, themselves form a model category =-=[7, 31]-=- such that algebras over cofibrant operads are homotopy invariant, see also [104]. Minimal operads mentioned in the previous item are particular cases of cofibrant dg-operads and the classical W-const... |

50 | Algebraic Geometry,” volume 52, Graduate Texts in Mathematics - Hartshorne |

50 |
Infinitesimal computations in topology
- Sullivan
- 1977
(Show Context)
Citation Context ...algebras over a reasonable (possibly colored) operad form a model category that generalizes the classical model structures of the categories of dg commutative associative algebras and dg Lie algebras =-=[95, 107]-=-. Operads, in asOPERADS AND PROPS 3 reasonable monoidal model category, themselves form a model category [7, 31] such that algebras over cofibrant operads are homotopy invariant, see also [104]. Minim... |

49 | On operad structures of moduli spaces and string theory
- Kimura, Stasheff, et al.
- 1995
(Show Context)
Citation Context ...fferential geometry [92, 93]. By the renaissance of operads we mean the first half of the nineties of the last century when several papers which stimulated the rebirth of interest in operads appeared =-=[31, 34, 41, 45, 47, 49, 72]-=-. Let us mention the most important new ideas that emerged during this period. Date: January 25, 2006. The author was supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Cze... |

44 |
Homotopy algebras are homotopy algebras
- Markl
- 2004
(Show Context)
Citation Context ...algebras, Chevalley-Eilenberg cohomology of Lie algebras, &c. Minimal models for operads [75] offer a conceptual understanding of strong homotopy algebras, their homomorphisms and homotopy invariance =-=[80]-=-. Operads serve as a natural language for various types of ‘multialgebras’ [64, 65]. Relation between Koszulness of operads and properties of posets was studied in [27]. Also the concept of the operad... |

39 |
homotopy algebra and iterated integrals for double loop spaces
- Operads
- 1994
(Show Context)
Citation Context ...fferential geometry [92, 93]. By the renaissance of operads we mean the first half of the nineties of the last century when several papers which stimulated the rebirth of interest in operads appeared =-=[31, 34, 41, 45, 47, 49, 72]-=-. Let us mention the most important new ideas that emerged during this period. Date: January 25, 2006. The author was supported by the grant GA ČR 201/05/2117 and by the Academy of Sciences of the Cze... |

37 |
Models for operads
- Markl
- 1996
(Show Context)
Citation Context ... of operads and deformations of their algebras was recognized. See [63] for an autochthonous account of the renaissance. Other papers that later became influential then followed in a rapid succession =-=[30, 33, 32, 73, 75]-=-. Let us list some most important outcomes of the renaissance of operads. The choice of the material for this incomplete catalog has been of course influenced by the author’s personal expertise and in... |

34 |
Homotopy Lie algebras
- Hinich, Schechtman
- 1993
(Show Context)
Citation Context |

32 | The Eckmann-Hilton argument, higher operads and En-spaces
- Batanin
(Show Context)
Citation Context ...d in [58] to mixed Tate motives over the rationals. See also an overview [88]. Category theory. Operads and multicategories were used as a language in which to propose a definition of weak ω-category =-=[3, 4, 5, 61]-=-. Operads themselves can be viewed as special kinds of algebraic theory (as can multicategories, if one allows many-sorted theories), see [85]. There are also ‘categorical’ generalizations of operads,... |

32 | Operads in higher-dimensional category theory
- Leinster
- 2000
(Show Context)
Citation Context ...more earlier, M. Boardman and R. Vogt [9] described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane [67]. As pointed out in =-=[62]-=-, also Lambek’s definition of multicategory [60] (late 1960s) was almost equivalent to what is called today a colored or many-sorted operad. Another important precursor was the associahedron K that ap... |

31 |
On perturbations and A∞-structures
- Gugenheim, Stasheff
- 1981
(Show Context)
Citation Context ... 2 Bt is isomorphic to the bialgebra PROP B. In other words, the PROP for bialgebras is a deformation of the PROP for 1bialgebras. According to general principles of homological perturbation 2 theory =-=[35]-=-, one may try to construct the resolution R as a perturbation of a cofibrant resolution 1 1 1 1 R of the PROP B. Since B is simpler that B, one may expect that resolving B would be a 2 2 2 2 simpler t... |

31 |
Algèbres ayant deux opérations associatives (digèbres
- Loday
- 1995
(Show Context)
Citation Context ...r operads [75] offer a conceptual understanding of strong homotopy algebras, their homomorphisms and homotopy invariance [80]. Operads serve as a natural language for various types of ‘multialgebras’ =-=[64, 65]-=-. Relation between Koszulness of operads and properties of posets was studied in [27]. Also the concept of the operadic distributive law turned out to be useful [26, 74]. Model structures. It turned o... |

30 | Operads and moduli spaces of genus 0 Riemann surfaces, The moduli space of curves (Texel Island
- Getzler
- 1994
(Show Context)
Citation Context ... of operads and deformations of their algebras was recognized. See [63] for an autochthonous account of the renaissance. Other papers that later became influential then followed in a rapid succession =-=[30, 33, 32, 73, 75]-=-. Let us list some most important outcomes of the renaissance of operads. The choice of the material for this incomplete catalog has been of course influenced by the author’s personal expertise and in... |

28 | Koszul duality for dioperads
- Gan
(Show Context)
Citation Context ...because the graph on the right hand side of the above display is not connected. Properads are still huge objects. The first really small version of PROPs were dioperads introduced in 2003 by W.L. Gan =-=[28]-=-. As a motivation for his definition, consider the following: Example 63. A Lie bialgebra is a vector space V with a Lie algebra structure [−, −] = : V ⊗ V → V and a Lie diagonal δ = : V → V ⊗ V . We ... |

27 | Cyclic operads and cyclic homology
- Getzler, Kapranov
- 1995
(Show Context)
Citation Context ... of operads and deformations of their algebras was recognized. See [63] for an autochthonous account of the renaissance. Other papers that later became influential then followed in a rapid succession =-=[30, 33, 32, 73, 75]-=-. Let us list some most important outcomes of the renaissance of operads. The choice of the material for this incomplete catalog has been of course influenced by the author’s personal expertise and in... |

27 | Vertex operator algebras and operads
- Huang, Lepowsky
- 1993
(Show Context)
Citation Context |

26 | Distributive laws and Koszulness - Markl - 1996 |

24 | Definitions: operads, algebras and modules
- May
- 1997
(Show Context)
Citation Context ... formulas involving maps in terms of elements, which is sometimes a welcome simplification. We believe that the reader can easily reformulate our definitions into other monoidal categories or consult =-=[83, 87]-=-. Let k[Σn] denote the k-group ring of the symmetric group Σn. Definition 1 (May’s operad). An operad in the category of k-modules is a collection P = {P(n)}n≥0 of right k[Σn]-modules, together with k... |

22 |
Homotopy coherent category theory and A∞-structures in monoidal categories
- Batanin
- 1998
(Show Context)
Citation Context ...d in [58] to mixed Tate motives over the rationals. See also an overview [88]. Category theory. Operads and multicategories were used as a language in which to propose a definition of weak ω-category =-=[3, 4, 5, 61]-=-. Operads themselves can be viewed as special kinds of algebraic theory (as can multicategories, if one allows many-sorted theories), see [85]. There are also ‘categorical’ generalizations of operads,... |

22 | Modern foundations of stable homotopy theory. This volume
- Elmendorf, Kriz, et al.
(Show Context)
Citation Context ... conjecture [44]. Topology. Operads as gadgets organizing homotopy coherent structures are important in the brave new algebra approach to topological Hochschild cohomology and algebraic K-theory, see =-=[22, 23, 90, 115]-=-, or [21] for a historical background. A description of a localized category of integral and p-adic homotopy types by E∞-operads was given in [69, 70]. An operadic approach to partial algebras and the... |

22 |
Cotangent cohomology of a category and deformations
- Markl
- 1996
(Show Context)
Citation Context |

22 | algebras and modules in general model categories
- Spitzweck, Operads
(Show Context)
Citation Context ...as [95, 107]. Operads, in asOPERADS AND PROPS 3 reasonable monoidal model category, themselves form a model category [7, 31] such that algebras over cofibrant operads are homotopy invariant, see also =-=[104]-=-. Minimal operads mentioned in the previous item are particular cases of cofibrant dg-operads and the classical W-construction [9] is a functorial cofibrant replacement in the category of topological ... |

21 | A Koszul duality for props
- Vallette
(Show Context)
Citation Context ...es, automorphisms of free groups or primitives in the homology of certain infinite-dimensional Lie algebras, see also [83, II.5.5]. In the same vein, complexes of directed graphs are related to PROPs =-=[84, 111, 112, 113]-=- and directed graphs with back-in-time edges are tied to wheeled PROPs introduced in [93]. Deformation theory and homotopy invariant structures in algebra. A concept of homotopy invariant structures i... |

20 |
Homotopy Associativity of H-spaces I,II
- Stasheff
- 1963
(Show Context)
Citation Context ...tegory [60] (late 1960s) was almost equivalent to what is called today a colored or many-sorted operad. Another important precursor was the associahedron K that appeared in J.D. Stasheff’s 1963 paper =-=[106]-=- on homotopy associativity of H-spaces. We do not, however, aspire to write an account on the history of operads and their applications here – we refer to the introduction of [83], to [89], [114], or ... |

20 | T-catégories (catégories dans un triple), Cahiers de Topologie et Géométrie Différentielle XII(3 - Burroni - 1971 |

18 |
Distributive laws, bialgebras, and cohomology Operads
- Fox, Markl
- 1995
(Show Context)
Citation Context ...g theory [49] and Connes-Kreimer’s approach to renormalization [15]. Operads and multicategories are important also for BeilinsonDrinfeld’s theory of chiral algebras [6]. Algebra. Operadic cohomology =-=[1, 26, 31, 34, 83]-=- provides a uniform treatment of all ‘classical’ cohomology theories, such as the Hochschild cohomology of associative algebras, Harrison cohomology of associative commutative algebras, Chevalley-Eile... |