## Axiomatic Homotopy Theory for Operads (2002)

Venue: | Comment. Math. Helv |

Citations: | 52 - 7 self |

### BibTeX

@ARTICLE{Berger02axiomatichomotopy,

author = {Clemens Berger and Ieke Moerdijk},

title = {Axiomatic Homotopy Theory for Operads},

journal = {Comment. Math. Helv},

year = {2002},

volume = {78},

pages = {020609--4}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

### Citations

327 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...1], Hinich [9], and others. Symmetric operads may be defined in any symmetric monoidal category. We propose here a general homotopy theory for such operads by means of Quillen closed model categories =-=[18]-=-. We show that under certain conditions the (reduced) operads of a symmetric monoidal model category carry a model structure with weak equivalences and fibrations defined on the level of the underlyin... |

319 | Homotopy associativity of H-spaces - Stasheff - 1963 |

230 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
(Show Context)
Citation Context ...rad generated by a collection. We discuss some preliminaries on the free functor F : Coll(E) ! Oper(E). The free functor is based on an operation known as grafting of trees, as explained e.g. in [7], =-=[8]-=- or [15]. To describe this grafting operation, we introduce the groupoid T of planar trees and non-planar isomorphisms: The objects of T are finite rooted planar trees. We closely follow the conventio... |

190 |
Homotopy invariant algebraic structures on topological. spaces
- Boardman, Vogt
(Show Context)
Citation Context ... up to a coherent system of homotopies. Stasheff's A1 -spaces resp. Boardman, Vogt and May's E1 -spaces are spaces with a "homotopy associative" resp. "homotopy commutative" multip=-=lication, cf. [23], [3]-=-, [16]. The corresponding A1 - resp. E1 -operad is a deformation of the operad acting on strictly associative resp. strictly commutative monoids. This method of deforming algebraic structures via oper... |

145 | Algebras and modules in monoidal model categories
- Schwede, Shipley
(Show Context)
Citation Context ...ed under pushouts along cofibrations. A sufficient condition for left properness is that all objects of the model category be cofibrant. 2.2. Monoidal model categories. A monoidal model category (cf. =-=[20]-=-) is a closed symmetric monoidal category endowed with a model structure subject to the following pushout-product axiom: For any pair of cofibrations f : X ae Y and f 0 : X 0 ae Y 0 , the induced map ... |

124 |
homotopy algebra and iterated integrals for double loop spaces arXiv:hep-th/9403055v1
- Getzler, Jones, et al.
(Show Context)
Citation Context ...\Omega P (n k ) ! P (n 1 + \Delta \Delta \Delta + n k ) satisfying well known equivariance, associativity and unit conditions. For more details, see e.g. Boardman-Vogt [3], May [16] and Getzler-Jones =-=[6]-=-. The category of operads in E is denoted by Oper(E). A cooperad is a collection C equipped with a counit C(1) ! I and structure maps C(n 1 +\Delta \Delta \Delta+n k ) ! C(k)\Omega C(n 1 )\Omega \Delt... |

114 | Model categories of diagram spectra
- Mandell, May, et al.
(Show Context)
Citation Context ...invariant of the p-adic homotopy type of X , provided that R is a field of characteristic p with surjective Frobenius map. 4.6.4. E1 -ring spectra. No monoidal model category for stable homotopy, cf. =-=[13]-=-, can simultaneously satisfy the first two hypotheses of (3.1), by a well known argument due to Lewis. However, all known models for stable homotopy are enriched either in simplicial sets or in topolo... |

81 | Homological algebra of homotopy algebras
- Hinich
- 1997
(Show Context)
Citation Context ...ictly associative resp. strictly commutative monoids. This method of deforming algebraic structures via operads has recently received new interest, cf. Mandell [16], Kontsevich-Soibelman [11], Hinich =-=[9]-=-, and others. Symmetric operads may be defined in any symmetric monoidal category. We propose here a general homotopy theory for such operads by means of Quillen closed model categories [18]. We show ... |

51 | Combinatorial operad actions on cochains
- Berger, Fresse
(Show Context)
Citation Context ..., the admissibility of cofibrant operads may be established using (4.1): all topological resp. simplicial operads are admissible by (4.1c). Cofibrant chain operads are admissible by a construction of =-=[2]-=-: indeed, in the category of R-chain operads there exists a \Sigma-cofibrant resolution E1 i Com together with a canonical E1 -coalgebra structure on the standard R-chain interval. For each R-chain op... |

37 |
Models for operads
- Markl
- 1996
(Show Context)
Citation Context ...he principal benefit of a model structure on operads is an intrinsic definition of homotopy algebras over an operad, namely as the algebras over a cofibrant replacement of the given operad, cf. Markl =-=[14]-=-. The algebras over cofibrant operads carry a model structure for which a variant of the BoardmanVogt homotopy invariance property holds. A larger class of operads, here called admissible \Sigma-cofib... |

30 |
Model categories and more general abstract homotopy theory, Book in preparation, available at http://www-math.mit.edu/ ∼ psh
- Dwyer, Hirschhorn, et al.
(Show Context)
Citation Context ...transfer of model structures. A model category is cofibrantly generated if the category is cocomplete and admits generating sets of (trivial) cofibrations with small domains, cf. DwyerHirschhorn -Kan =-=[5] and Hovey [10], who-=- use a slightly more general concept. "Small" means -small for some regular cardinalsand "generating" means that the fibrations (resp. trivial fibrations) are characterized by thei... |

27 | Cyclic operads and cyclic homology
- Getzler, Kapranov
- 1995
(Show Context)
Citation Context ...e operad generated by a collection. We discuss some preliminaries on the free functor F : Coll(E) ! Oper(E). The free functor is based on an operation known as grafting of trees, as explained e.g. in =-=[7]-=-, [8] or [15]. To describe this grafting operation, we introduce the groupoid T of planar trees and non-planar isomorphisms: The objects of T are finite rooted planar trees. We closely follow the conv... |

26 | Operads, homotopy algebras and iterated integrals for double loop spaces
- Getzler, Jones
- 1994
(Show Context)
Citation Context ...s n≥0 P(k) ⊗ P(n1) ⊗ · · · ⊗ P(nk) → P(n1 + · · · + nk) satisfying well known equivariance, associativity and unit conditions. For more details, see e.g. Boardman-Vogt [3], May [16] and Getzler-Jones =-=[6]-=-. The category of operads in E is denoted by Oper(E). A cooperad is a collection C equipped with a counit C(1) → I and structure maps C(n1+· · ·+nk) → C(k)⊗C(n1)⊗· · ·⊗C(nk) satisfying the dual condit... |

24 | 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 ...easy to verify, and are well known to hold in many standard situations. In particular, they hold for topological, simplicial and chain operads. Our approach may be compared with Hinich [9], Spitzweck =-=[22]-=- and Vogt [24], but is much more elementary, since it relies on the existence of path-objects rather than on an intricate analysis of pushouts. Our method also immediately extends to coloured operads.... |

20 |
A 1 -homotopy theory of schemes, Inst
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...valences" as weak equivalences. In the simplicial case, the pushout-product axiom is easy to verify; in the R-chain-complex case, the pushout-product axiom only holds if R is a field. Morel-Voevo=-=dsky [17]-=- show that the category of simplicial sheaves on a site T of finite type admits a symmetric monoidal fibrant replacement functor. If R is a field, such a replacement functor can also be constructed fo... |

13 |
E1 algebras and p–adic homotopy theory, Topology 40
- Mandell
- 2001
(Show Context)
Citation Context ...r dually, a E1 -algebra structure on the cochains) of any simplicial set. It follows from (4.1a), that E1 is admissible, i.e. an E1 -operad for the category of R-chain complexes. According to Mandell =-=[12]-=-, an E1 - structure on the cochains of a nilpotent simplicial set X is a complete invariant of the p-adic homotopy type of X , provided that R is a field of characteristic p with surjective Frobenius ... |

12 |
The geometry of iterated loop spaces
- P
(Show Context)
Citation Context ...o a coherent system of homotopies. Stasheff's A1 -spaces resp. Boardman, Vogt and May's E1 -spaces are spaces with a "homotopy associative" resp. "homotopy commutative" multiplicat=-=ion, cf. [23], [3], [16]-=-. The corresponding A1 - resp. E1 -operad is a deformation of the operad acting on strictly associative resp. strictly commutative monoids. This method of deforming algebraic structures via operads ha... |

11 |
Homotopy theory of coalgebras
- Smirnov
(Show Context)
Citation Context ...algebra structure on A corresponds also to an operad map P ! EA with values in the endomorphism-operad, defined by EA (n) = A (A\Omega n ) with the natural compositional operad structure, cf. Smirnov =-=[21]-=-. Dually, a P -coalgebra structure on B corresponds to an operad map P ! E op B with values in the coendomorphism-operad, defined by E op B (n) = (B\Omega n ) B . The product P\Omega Q of two operads ... |

11 |
Homotopy associativity of H-spaces, Trans
- Stasheff
- 1963
(Show Context)
Citation Context ...g only up to a coherent system of homotopies. Stasheff's A1 -spaces resp. Boardman, Vogt and May's E1 -spaces are spaces with a "homotopy associative" resp. "homotopy commutative" =-=multiplication, cf. [23]-=-, [3], [16]. The corresponding A1 - resp. E1 -operad is a deformation of the operad acting on strictly associative resp. strictly commutative monoids. This method of deforming algebraic structures via... |

11 |
E∞-algebras and p-adic homotopy theory, Topology 40
- Mandell
- 2001
(Show Context)
Citation Context ...(or dually, a E∞-algebra structure on the cochains) of any simplicial set. It follows from (4.1a), that E∞ is admissible, i.e. an E∞-operad for the category of R-chain complexes. According to Mandell =-=[12]-=-, an E∞structure on the cochains of a nilpotent simplicial set X is a complete invariant of the p-adic homotopy type of X, provided that R is a field of characteristic p with surjective Frobenius map.... |

11 | The Boardman-Vogt resolution of operads in monoidal model categories, in preparation - Berger, Moerdijk |

5 | Crans – Quillen closed model structures for sheaves - E - 1995 |

5 | Every homotopy theory of simplicial algebras admits a proper model
- Rezk
(Show Context)
Citation Context ...ce the projective generators represent evaluation at n, Quillen’s structure coincides with ours. There is another model structure on simplicial operads, obtained by restricting Rezk’s model structure =-=[19]-=- on simplicial theories to simplicial operads. The class of weak equivalences for this model structure is smaller: in particular, two simplicial operads are weakly equivalent if and only if they defin... |

3 |
Soibelmann – Deformations of algebras over operads and Deligne’s
- Kontsevich, Y
(Show Context)
Citation Context ...acting on strictly associative resp. strictly commutative monoids. This method of deforming algebraic structures via operads has recently received new interest, cf. Mandell [16], Kontsevich-Soibelman =-=[11]-=-, Hinich [9], and others. Symmetric operads may be defined in any symmetric monoidal category. We propose here a general homotopy theory for such operads by means of Quillen closed model categories [1... |

1 | Model categories, Surveys and - Hovey - 1999 |

1 |
Cofibrant operads and universal E1 -operads, Bielefeld SB 343
- Vogt
- 1999
(Show Context)
Citation Context ..., and are well known to hold in many standard situations. In particular, they hold for topological, simplicial and chain operads. Our approach may be compared with Hinich [9], Spitzweck [22] and Vogt =-=[24]-=-, but is much more elementary, since it relies on the existence of path-objects rather than on an intricate analysis of pushouts. Our method also immediately extends to coloured operads. The principal... |

1 |
Cofibrant operads and universal E∞-operads, Bielefeld SB 343
- Vogt
- 1999
(Show Context)
Citation Context ..., and are well known to hold in many standard situations. In particular, they hold for topological, simplicial and chain operads. Our approach may be compared with Hinich [9], Spitzweck [22] and Vogt =-=[24]-=-, but is much more elementary, since it relies on the existence of path-objects rather than on an intricate analysis of pushouts. Our method also immediately extends to coloured operads. The principal... |

1 | The Geometry of Iterated Loop Spaces, Lect - P |