## Operads, homotopy algebra, and iterated integrals for double loop spaces (1995)

Venue: | 15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146 |

Citations: | 26 - 0 self |

### BibTeX

@INPROCEEDINGS{Getzler95operads,homotopy,

author = {E. Getzler and J. D. S. Jones},

title = {Operads, homotopy algebra, and iterated integrals for double loop spaces },

booktitle = {15 T. KASHIWABARA – ON THE HOMOTOPY TYPE OF CONFIGURATION COMPLEXES, CONTEMP. MATH. 146},

year = {1995},

pages = {159--170},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Chen's theory of iterated integrals provides a remarkable model for the differential forms on the based loop space M of a differentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of differentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding two-dimensional topological field theories. Our approach is to use the formalism of operads. Operads can be defined in any symmetric monoidal category, although we will mainly be concerned with dg-operads (differential graded operads), that is, operads in the category of chain complexes with monoidal structure defined by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative | we give a precise definition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a non-linear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the k-th tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or a-algebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)

### Citations

474 |
Categories for the working mathematician
- MacLane
- 1998
(Show Context)
Citation Context ....S. JONES In fact, any triple may be factored as T = F U, where U is left adjoint to F : this is proved by introducing the category of algebras over the triple, following Eilenberg-Moore (see MacLane =-=[32]-=-, Section VI.2). De nition 1.2. An algebra over a triple (T; ; ) is a pair (X; ) where X is an object of the category C and : TX �! X is a morphism, such that the composition X X ��! TX �! X is the id... |

327 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...sociated operads leaves out half the story, since one cannot then formulate theorems involving morphisms of operads. Many of our results are expressed in the language of Quillen's homotopical algebra =-=[40]-=-. This is a non-linear generalization of homological algebra, allowing the construction of derived functors in categories with some of the structure of homotopy theory. Such categories are called clos... |

269 |
On the structure of Hopf algebras
- Milnor, Moore
- 1965
(Show Context)
Citation Context ... 1 �! A: these two spaces are naturally isomorphic, since " : 1 �! 1 is the identity. Denote by the product and by the coproduct of A. There are two ltrations on any bialgebra A (see Milnor and Moore =-=[35]-=- and Quillen [42]). (1) The descending ltration F i A, where F 0 A = A and F i A is the i-th power of A if i 1. (2) The ascending ltration FiA, where F0A = 0 and FiA is the kernel of the iterated copr... |

260 | The geometry of iterated loop spaces
- May
- 1972
(Show Context)
Citation Context ...tually specialize to two categories. (1) Let T be the category of compactly generated topological spaces, with tensor product the product of spaces. An operad in T is called a topological operad (May =-=[33]-=-): this generalizes the notion of a topological monoid. (2) Let M be the category of chain complexes V over a eld K with Vi = 0 for i < 0, and with tensor product the graded tensor product. An operad ... |

227 |
The cohomology structure of an associative ring
- Gerstenhaber
- 1963
(Show Context)
Citation Context ...iven by the formula (c)[a1j : : :jak] = X 0 i<j k (�1) (jcj+1)(ja1j+ +jaij+i) [a1j : : :jaijc[ai+1j : : :jaj]jaj+1j : : :jak]:OPERADS AND HOMOTOPY ALGEBRA 49 Note that j (c)j = jcj + 1. Gerstenhaber =-=[16]-=- has introduced an operation c1 c2 of degree 1 on C (V; V ): (c1 c2)[a1j : : :jak] = X 0 i<j k (�1) (jc2j+1)(ja1j+ +jaij+i) c1[a1j : : :jaijc2[ai+1j : : :jaj]jaj+1j : : :jak]: The commutator [c1; c2] ... |

224 |
Rational homotopy theory
- Quillen
- 1969
(Show Context)
Citation Context ...omotopy equivalences between the categories of associative dg-algebras and connected coassociative dg-coalgebras, and between the categories of commutative dg-algebras and connected Lie dg-coalgebras =-=[42]-=-. Our strategy for proving the above theorem is to generalize work of Stashe [49]. He introduced a sequence of convex (k � 2)-dimensional polyhedra K(k), the associahedra, such that the products K(k) ... |

190 |
Homotopy invariant algebraic structures on topological. spaces
- Boardman, Vogt
(Show Context)
Citation Context ...y respectively. One of the main applications of topological operads is to the study of iterated loop spaces; for two excellent overviews of this subject, see Adams [2] and May [34]. Boardman and Vogt =-=[9]-=- introduced a sequence of operads, the little n-cubes operads Fn(k), which have the homotopy type of the con guration space Fn(k) of k distinct points in Rn . A space with an action of the operad Fn i... |

189 | Toposes, Triples, and Theories
- Barr, Wells
- 1985
(Show Context)
Citation Context ...section, we will show that the category of operads has all limits and colimits. This follows from the following theorem, which is a rather special case of the results of Section 9.3 of Barr and Wells =-=[6]-=-. Theorem 1.13. Let C be a category with all limits and colimits, and let T : C �! C be a triple which preserves ltered colimits. Then the category of T-algebras C T has all limits and colimits. The d... |

140 |
Formal Noncommutative Symplectic Geometry
- Kontsevich
- 1993
(Show Context)
Citation Context ... the rst three sections of this chapter, we study three operads: A1, introduced by Stashe [49], B1, which we abstract from the work of Baues [7], and C1, considered by Kadeishvili [27] and Kontsevich =-=[29]-=-. We de ne these operads in a uniform way: an algebra over one of these operads is a chain complex A together with additional algebraic structure on the free coalgebra generated by A: for A1-algebras,... |

131 |
Arrangements of hyperplanes and Lie algebra homology
- Schechtman, Varchenko
- 1991
(Show Context)
Citation Context ...care-Birkho -Witt theorem, while for n > 1, it is an immediate consequence of Theorem 1.6. An entirely di erent approach to Theorem 1.6 when n = 2 is contained in the work of Schechtman and Varchenko =-=[45]-=-: they prove the more general result, that if S is a nite subset of the complex plane, then 1M k=0 H (FCnS(k)) Sk V (k) = T + 1( �1 TL( V )) T + 1 ( V )(S) : It also follows from Theorem 1.6 that for ... |

127 | Introduction to sh Lie Algebras for Physicists, Int
- Lada, Stasheff
- 1993
(Show Context)
Citation Context ...bra on A for the almost free resolution B a ? of a. We show that for the Lie operad L, we recover the notion of a homotopy Lie algebra of Drinfeld [12], Hinich and Schechtman [25] and Lada and Stashe =-=[30]-=-. All operads in this section will be exact and augmented, and all maps of operads will respect the augmentation. All cooperads will be connected, exact and coaugmented. 4.1. Closed model categories. ... |

124 | Chern–Simons perturbation theory
- Axelrod, Singer
- 1994
(Show Context)
Citation Context ...er M nX and the normal sphere bundle of X: this is once more a manifold with corners. (For a more detailed description of the Fulton-MacPherson compacti cation in this context, see Axelrod and Singer =-=[4]-=-).@ @ @ � r � � @ @ � @ r � @ � r � @ @ @ @ r � @ � r � @ � � @ r � � @ @ � @ r � @ � r � 34 E. GETZLER AND J.D.S. JONES If S is a nite set, let Bl ((R n ) S ) be the blow-up of (R n ) S along the di... |

122 | Batalin-Vilkovisky algebras and two-dimensional topological field theories
- Getzler
- 1994
(Show Context)
Citation Context ...mation theory of homotopy Batalin-Vilkovisky algebras (which are closely related to 2-algebras) is central to string eld theory, or at least to its genus zero approximation (Zwiebach [51] and Getzler =-=[19]-=-). In particular, the notion of a homotopy Batalin-Vilkovisky algebra, analogous to the notion of homotopy 2-algebra of Section 4.4, is precisely the algebraic structure induced by genus zero correlat... |

79 |
Foncteurs analytiques et espèces de structures
- Joyal
- 1985
(Show Context)
Citation Context ...ory of S-objects, or functors from Sto C, is a monoidal category with respect to a certain tensor product, and operads are monoids in this monoidal category. (This tensor product was studied by Joyal =-=[26]-=-, who calls S-objects species, and by Smirnov [46].) Representations of operads are called algebras: for example, associative and commutative algebras are algebras over certain operads e1 and e1 in th... |

74 |
Infinite loop spaces
- Adams
- 1978
(Show Context)
Citation Context ...d T = U T F T . 1.2. Operads. From now on, we will restrict attention to a category C with the following properties: (1) C is a symmetric monoidal category with tensor product functor � � and unit 1; =-=(2)-=- C has all small limits and colimits; (3) for any object X, the functor X � preserves colimits. Denote by 0 the initial object of C, which exists by the assumption that C has colimits. We will eventua... |

70 |
A∞-algebras and the cyclic bar complex
- GETZLER, JONES
- 1989
(Show Context)
Citation Context ...vation of BV of degree �1 is a codi erential if and only if the corresponding Hochschild cochain m (of degree �2) satis es the formula m m = 0. The following de nition is due to Stashe [49] (see also =-=[20]-=-). De nition 5.1. An A1-algebra structure on a graded vector space A is one of the two equivalent data: (1) a Hochschild cochain m 2 Hom(BA; A) of degree �2 such that m m = 0; (2) a codi erential on B... |

62 |
Lie elements and an algebra associated with shuffles
- Ree
- 1958
(Show Context)
Citation Context ...rison cochain, it is easily seen that the associated coderivation (c) is a derivation with respect to the shu e map.OPERADS AND HOMOTOPY ALGEBRA 53 T + 1 Ree has studied the bialgebra U dual to ours =-=[43]-=-. Its underlying algebra is the tensor algebra (V ) of a connected vector space V , with coproduct the sum over unshu es X n�1 (a1 : : : an) = k=1 X 2S(k;n�k) (a �1 (1) : : : a �1 (k)) (a �1 (k+1) : :... |

59 |
The Cohomology Ring of the Colored Braid Group
- Arnold
(Show Context)
Citation Context ..., we will restrict attention to a category C with the following properties: (1) C is a symmetric monoidal category with tensor product functor � � and unit 1; (2) C has all small limits and colimits; =-=(3)-=- for any object X, the functor X � preserves colimits. Denote by 0 the initial object of C, which exists by the assumption that C has colimits. We will eventually specialize to two categories. (1) Let... |

59 |
On the (co-) homology of commutative rings
- Quillen
- 1968
(Show Context)
Citation Context ...ts ai have even degree. The commutativity of a homotopy commutative algebra is re ected by the fact that the di erential commutes with the antipode. We may now prove a theorem of Barr [5] and Quillen =-=[41]-=-, which identi es the Harrison homology of a commutative algebra in characteristic zero with its Andre-Quillen homology. Theorem 5.8. Let A be a commutative algebra over a homotopy equivalence of comp... |

53 |
Knot groups
- Neuwirth
(Show Context)
Citation Context ...n embedding, n is a total order on S. We may now decompose the con guration space Fn(S) into convex cells. (In the special case n = 2, the decomposition which we describe may be found in Fox-Neuwirth =-=[14]-=-, though the description of the boundary given there is incorrect.) Denote by ( 1 n) Map(S;Rn ) the convex set consisting of those maps f : S �! Rn such that the induced ag of preorders on S is 1 n. T... |

46 |
homotopy formulas and the Gauss-Manin connection in cyclic homology
- GETZLER, Cartan
- 1993
(Show Context)
Citation Context ...y associative algebras in the sense of Section 4.4. In the next section, we will need a sequence of operations c0fc1; : : :; ckg on the space of Hochschild cochains of a chain complex, constructed in =-=[18]-=-, generalizing Gerstenhaber's operation c0fc1g = c1, given by the formula c0 (1) cfc1; : : : ; ckg[a1j : : :ja`] = X (�1) i 1 jc1j+ + i k jckj 0 i1 j1 ik jk ` c[a1j : : :jai1jc1[ai1+1j : : :jaj1]jaj1+... |

45 |
The homology of Cn+1-spaces, n ≥ 0. In The homology of iterated loop spaces., volume 533
- Cohen
- 1976
(Show Context)
Citation Context ...e dg-algebras, while e1-algebras are commutative dg-algebras. We adopt the term n-algebra for an en-algebra. For n > 1, the structure of n-algebras was explicitly determined by F. Cohen in his thesis =-=[11]-=-. An n-algebra is a commutative dg-algebra A with a graded Lie bracket [a; b] of degree n � 1 (that is, a graded Lie bracket on the (n � 1)-fold suspension n�1A of A) satisfying the Poisson relation [... |

40 |
On the cobar construction
- Adams
- 1956
(Show Context)
Citation Context ...tudy in Chapter 5: in the special case of n = 1, this is just the operad A1. Let S (X) be the dg-algebra of singular cochains on a topological space X, with integral coe cients. By a theorem of Adams =-=[1]-=-, if X is simply connected there is a natural isomorphism in homology between the bar construction BS (X) and S ( X). Many authors have attempted to nd an analogue of this theorem for iterated loop sp... |

35 |
Opération sur l'homologie cyclique des algèbres commutatives
- Loday
- 1989
(Show Context)
Citation Context ...uer [44]). The induced idempotents on the bar coalgebra BA commute with the Hochschild di erential if A is a C1-algebra, and we obtain the Hodge decomposition of BA (GerstenhaberSchack [17] and Loday =-=[31]-=-). The simplest of these idempotents, which Barr constructed by hand, projects from the Hochschild complex to the Harrison complex, permitting us to show that the Harrison complex of a free commutativ... |

34 |
Homotopy Lie algebras
- Hinich, Schechtman
- 1993
(Show Context)
Citation Context ... the structure of an algebra on A for the almost free resolution B a ? of a. We show that for the Lie operad L, we recover the notion of a homotopy Lie algebra of Drinfeld [12], Hinich and Schechtman =-=[25]-=- and Lada and Stashe [30]. All operads in this section will be exact and augmented, and all maps of operads will respect the augmentation. All cooperads will be connected, exact and coaugmented. 4.1. ... |

27 | Infinitesimal structure of moduli spaces of G-bundles, Internat - Beilinson, Ginzburg - 1992 |

26 |
The action of Sn on the components of the Hodge decomposition of Hochschild homology
- Hanlon
- 1990
(Show Context)
Citation Context ...F 1z1=F 2z1. Over a eld of characteristic zero, the Poincare-Birkho -Witt theorem shows the existence of a family of commuting idempotents on z1 splitting this ltration (Solomon [48]; see also Hanlon =-=[24]-=- and Reutenauer [44]). The induced idempotents on the bar coalgebra BA commute with the Hochschild di erential if A is a C1-algebra, and we obtain the Hodge decomposition of BA (GerstenhaberSchack [17... |

19 |
homology, Hochschild homology and triples
- Barr, Harrison
- 1968
(Show Context)
Citation Context ...ll of the elements ai have even degree. The commutativity of a homotopy commutative algebra is re ected by the fact that the di erential commutes with the antipode. We may now prove a theorem of Barr =-=[5]-=- and Quillen [41], which identi es the Harrison homology of a commutative algebra in characteristic zero with its Andre-Quillen homology. Theorem 5.8. Let A be a commutative algebra over a homotopy eq... |

17 |
On the groups H(#, n
- Eilenberg, MacLane
- 1953
(Show Context)
Citation Context ...tor B n 1 = �n B n from commutative di erential graded algebras to chain complexes obtained by iterating this functor n times is central to Eilenberg and MacLane's approach to the homology of K( ; n) =-=[13]-=-. The following result follows straightforwardly from our description of the boundary in Zn. Proposition 5.14. There is a natural equivalence of endofunctors on the category of chain complexes T(Zn; V... |

15 |
The shuffle bialgebra and the cohomology of commutative algebras
- Gerstenhaber, Schack
- 1991
(Show Context)
Citation Context ...24] and Reutenauer [44]). The induced idempotents on the bar coalgebra BA commute with the Hochschild di erential if A is a C1-algebra, and we obtain the Hodge decomposition of BA (GerstenhaberSchack =-=[17]-=- and Loday [31]). The simplest of these idempotents, which Barr constructed by hand, projects from the Hochschild complex to the Harrison complex, permitting us to show that the Harrison complex of a ... |

13 |
Algèbre homologique et homologie des espaces classifiants, Séminaire Cartan
- Moore
- 1959
(Show Context)
Citation Context ...f Eilenberg-MacLane if a is an algebra. If v is an S-module (an S-object in the category of chain complexes M), denote by v # the S-module with zero di erential underlying v (this is Moore's notation =-=[36]-=-). De nition 2.1. An operad a is almost free if the operad a # is free. A cooperad z is almost cofree if the cooperad z # is cofree.@ @ A @ A A @ � r @ � A A @ � A � r � � - @ @ A @ A A @ A r � � � �... |

11 |
A(∞)-algebra structure in cohomology and the rational homotopy type
- Kadeishvili
(Show Context)
Citation Context ...motopy n-algebras In the rst three sections of this chapter, we study three operads: A1, introduced by Stashe [49], B1, which we abstract from the work of Baues [7], and C1, considered by Kadeishvili =-=[27]-=- and Kontsevich [29]. We de ne these operads in a uniform way: an algebra over one of these operads is a chain complex A together with additional algebraic structure on the free coalgebra generated by... |

10 |
Homotopy theory of coalgebras, Izv
- Smirnov
- 1985
(Show Context)
Citation Context ...here S (X) is considered as an En-algebra, and �nS ( nX). For a di erent extension of Adam's theorem which uses simplicial methods and thus avoids the introduction of almost free operads, see Smirnov =-=[46]-=-, [47]. In Chapter 6, we examine the same questions from the point of view of rational homotopy and iterated integrals. Let A (M) be the dg-algebra of di erential forms on a smooth manifold M, negativ... |

8 |
Stasheff – Homotopy associativity of H-spaces
- D
- 1963
(Show Context)
Citation Context ...action of the operad Fn is called an En-space, and a connected En-space has the homotopy type of an n-fold loop space. Certain cases are of particular interest: n = 1 recovers the theory of A1-spaces =-=[49]-=-, n = 1 leads to in nite loop spaces, and n = 2 is intimately related to the braid groups B k, since F2(k)=Sk ' K(B k; 1). The homology en(k) = H (Fn(k); K) of the topological operad Fn over a eld K o... |

5 |
Koszul duality for operads, preprint
- Ginzburg, Kapranov
- 1993
(Show Context)
Citation Context ...Fn(k), and an application of Lefschetz duality. This method of proof was inspired by an article of BeilinsonGinburg [8]; in e ect, they considered the case n = 1. In the language of Ginzburg-Kapranov =-=[22]-=-, we show that the operad en is Koszul. In our exposition, we make use of their bar construction for dg-operads, which we view as a functor from dg-operads to dg-cooperads. We also make use the free o... |

4 | The life and work of Kuo-Tsai Chen - Hain, Tondeur - 1990 |

4 | On the chain complex of an iterated loop space, Izv - Smirnov - 1989 |

3 |
Letter to Schechtman
- Drinfeld
- 1988
(Show Context)
Citation Context ... C(a ?; A), or equivalently, the structure of an algebra on A for the almost free resolution B a ? of a. We show that for the Lie operad L, we recover the notion of a homotopy Lie algebra of Drinfeld =-=[12]-=-, Hinich and Schechtman [25] and Lada and Stashe [30]. All operads in this section will be exact and augmented, and all maps of operads will respect the augmentation. All cooperads will be connected, ... |

2 |
Petrack S.: Dierential forms on loop spaces and the cyclic bar complex
- Getzler, Jones
- 1991
(Show Context)
Citation Context ...ry of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack =-=[21]-=-). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic fra... |

2 |
The projectivity of moduli spaces of stable n-pointed curves
- Knudsen
- 1983
(Show Context)
Citation Context ... Stashe [49] in the study of homotopy associative spaces. (2) The space F2(k) is a circle bundle over the moduli space M0;k+1 of stable rational curves with k + 1 marked points constructed by Knudsen =-=[28]-=-. We may also describe Fn(2) and Fn(3) for general n. (1) The space Fn(2) = Fn(2) is the sphere S n�1 , re ecting the fact that there is only one tree with two leaves: 1 2 The group GL(n) acts by rota... |

2 |
DGA algebras as a Quillen model category. Relations to shm maps
- Munkholm
- 1978
(Show Context)
Citation Context ...odel categories; this generalizes the closed model categories of commutative dg-algebras over Q, Lie dg-algebras over Q (Section II.5, Quillen [42]) and associative dg-algebras over any eld (Munkholm =-=[38]-=-): In Section 4.3, we prove that the adjoint pair of functors (a) and B(a) de nes an equivalence between the homotopy categories of a-operads and Ba-cooperads. We also study the total left derived fun... |

2 |
Theorem of Poincare-Birkho -Witt and symmetric group representations of degrees equal to the Stirling numbers
- Reutenauer
- 1986
(Show Context)
Citation Context ...eld of characteristic zero, the Poincare-Birkho -Witt theorem shows the existence of a family of commuting idempotents on z1 splitting this ltration (Solomon [48]; see also Hanlon [24] and Reutenauer =-=[44]-=-). The induced idempotents on the bar coalgebra BA commute with the Hochschild di erential if A is a C1-algebra, and we obtain the Hodge decomposition of BA (GerstenhaberSchack [17] and Loday [31]). T... |

2 |
On the Poincaré-Birkho -Witt theorem
- Solomon
- 1968
(Show Context)
Citation Context ...rem identi es z1 with F 1z1=F 2z1. Over a eld of characteristic zero, the Poincare-Birkho -Witt theorem shows the existence of a family of commuting idempotents on z1 splitting this ltration (Solomon =-=[48]-=-; see also Hanlon [24] and Reutenauer [44]). The induced idempotents on the bar coalgebra BA commute with the Hochschild di erential if A is a C1-algebra, and we obtain the Hodge decomposition of BA (... |

2 |
Closed string £eld theory: Quantum action and the Batalin-Vilkovisky master equation
- Zwiebach
- 1993
(Show Context)
Citation Context ...ed that the deformation theory of homotopy Batalin-Vilkovisky algebras (which are closely related to 2-algebras) is central to string eld theory, or at least to its genus zero approximation (Zwiebach =-=[51]-=- and Getzler [19]). In particular, the notion of a homotopy Batalin-Vilkovisky algebra, analogous to the notion of homotopy 2-algebra of Section 4.4, is precisely the algebraic structure induced by ge... |

1 |
The double bar annd cobar constructions
- Baues
- 1981
(Show Context)
Citation Context ...even twice, since S (X) is not a commutative dg-algebra, and thus it is not evident that BS (X) has an associative product. For further iterations, the problem only becomes worse. Nevertheless, Baues =-=[7]-=- has constructed an associative product on BS (X) using Steenrod's [1 operation and certain multilinear analogues, which allow him to construct the structure of a dg-bialgebra on BS (X). He shows that... |

1 |
A compacti cation of con guration spaces, to appear
- Fulton, MacPherson
(Show Context)
Citation Context ...hich is the case n = 1. The construction of Fn(k) makes use of a compacti cation of the con guration space Fn(k), or rather, its quotient by translations and dilatations, due to Fulton and MacPherson =-=[15]-=-. The operad Fn is obtained by gluing together faces corresponding to components of the free operad generated by the top-dimensional strata of the spaces Fn. (In a sense which we do not make precise i... |

1 |
nite loop space theory
- May, In
- 1977
(Show Context)
Citation Context ... and Lie algebra homology respectively. One of the main applications of topological operads is to the study of iterated loop spaces; for two excellent overviews of this subject, see Adams [2] and May =-=[34]-=-. Boardman and Vogt [9] introduced a sequence of operads, the little n-cubes operads Fn(k), which have the homotopy type of the con guration space Fn(k) of k distinct points in Rn . A space with an ac... |

1 |
erential homological algebra, Actes du Congres International des Mathematiciens
- Moore, Di
- 1970
(Show Context)
Citation Context ...tion are weak equivalences. Thus, the operad B Ba may be thought of as a canonical almost free resolution of the operad a. The analogous construction for algebras is well-known; see for example Moore =-=[37]-=-. 2.2. Almost-free algebras and di erentials. The following de nition may be compared to that of an almost free operad of the last section. De nition 2.4. If a is an operad, an a-algebra A is almost f... |