## Contents

### BibTeX

@MISC{Vallette_contents,

author = {Bruno Vallette},

title = {Contents},

year = {}

}

### OpenURL

### Abstract

Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free

### Citations

244 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1993
(Show Context)
Citation Context ...ch that the associated categories of models are the same. The (co)homology theories and the lax notion “up to homotopy” of a particular type of (bi)algebras are given by the Koszul duality of operads =-=[GK]-=-, dioperads [G] or properads [V, MeVa]. To generalize Koszul duality theory from associative algebras [P] to operads, dioperads and properads, the first step is to extend to notions of bar and cobar c... |

202 |
Topos Theory
- JOHNSTONE
- 1977
(Show Context)
Citation Context ...ication functors preserve reflexive coequalizers, which is a weaker version of the notion of cokernel. For more details about reflexive coequalizers, we refer the reader to the book of P.T. Johnstone =-=[J]-=-. Definition (Reflexive coequalizer). A pair of morphisms X1 d1 d0 � X0 is said to be reflexive is there exists a morphism s0 : X0 → X1 such that d0◦s0 = d1◦s0 = idX0. A coequalizer of a reflexive pai... |

185 |
Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ...LETTE of n-level connected graphs by the relation V ⊠c I ≃ I ⊠c V , which is equivalent to forget the levels. □ The notion of properad is a “connected” version of the notion of prop (see F.W. Lawvere =-=[La]-=-, S. Mac Lane [MacL2] and J.F. Adams [A]). For more details about the link between these two notions we refer the reader to [V]. From the previous theorem, one can get the description of the free prop... |

143 |
Categories for the working mathematician”, second edition
- Lane
- 1997
(Show Context)
Citation Context ... In a monoidal category with denumerable coproducts, when the monoidal product preserves coproducts, the free monoid on an object V is well understood and is given by the words with letters in V (see =-=[MacL1]-=- Chapter VII Section 3 Theorem 2). In general, the existence of the free monoid has been established, under some hypotheses, by M. Barr in [B]. When the monoidal product preserves colimits over the si... |

51 |
A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so
- Kelly
- 1980
(Show Context)
Citation Context ...n [B]. When the monoidal product preserves colimits over the simplicial category, E. Dubuc described in [D] a construction for the free monoid. A general categorical answer was given by G.M. Kelly in =-=[K]-=- when the monoidal product preserves colimit on one side. Once again, its construction requires the tensor product to preserve colimits. The problem is that the monoidal products that appeared recentl... |

29 |
Spaces of algebra structures and cohomology of operads
- Rezk
- 1996
(Show Context)
Citation Context ...e side, G.M. Kelly gave a construction of the free monoid by means of a particular colimit in Equation (23.2) page 69 of [K] (see also H.J. Baues, M. Jibladze, A.Tonks in [BJT] Appendix B and C. Rezk =-=[R]-=- Appendix A). This construction applies to operads. Since the other monoidal products considered here do not preserve coproducts neither general colimits, we need to refine the arguments. For monoidal... |

28 | Koszul duality for dioperads
- Gan
(Show Context)
Citation Context ...over a monoid in a monoidal category. Moreover, in simple cases, one does not need the full machinery of props. For instance, Frobenius bialgebras and Lie bialgebras can be modelled by dioperads (see =-=[G]-=-) whereas associative bialgebras and involutive Lie bialgebras require the notion of properads (see [V]). These two notions are monoids which generate bigger props but such that the associated categor... |

25 |
André-Quillen (co)-homology for simplicial algebras over simplicial operads. In Une dégustation topologique [Topological morsels]: homotopy theory
- Goerss, Hopkins
- 1999
(Show Context)
Citation Context ...eserve reflexive coequalizers, which are is weaker version of the notion of cokernel. For more details about reflexive coequalizers, we refer the reader to the article of P.G. Goerss and M.J. Hopkins =-=[GH]-=-. Definition (Reflexive coequalizer). A pair of morphisms X1 d1 d0 � X0 is said to be reflexive is there exists un morphism s0 : X0 → X1 such that d0 ◦s0 = d1 ◦s0 = idX0. A coequalizer of a reflexive ... |

21 | private communication - KONTSEVICH |

17 | A resolution (minimal model) of the PROP for bialgebras
- Markl
(Show Context)
Citation Context ... Frobenuis algebras, infinitesimal bialgebras can be modelled by a dioperad (see [G]). 15BRUNO VALLETTE 5.4. Free special prop. In order to give the resolution of the prop of bialgebras, M. Markl in =-=[Ma]-=- defined the notion of special props. It is corresponds to monoids in the monoidal category of S-bimodules where the monoidal product is based only on composition called fractions ([Ma] definition 19)... |

16 | Resolution of coloured operads and rectification of homotopy algebras Categories in algebra, geometry and mathematical
- Berger, Moerdijk
- 2007
(Show Context)
Citation Context ...he colors of the operation below. C. Berger and I. Moerdijk defined a monoidal product of the category of colored collections such that the related monoids are exactly colored operad (see Appendix of =-=[BM]-=-). Once again, Theorem 10 applies in this case and we get the description of the free colored operad by means of trees without levels. Acknowledgements I would like to thank Michael Batanin, Benoit Fr... |

16 |
PROPped up graph cohomology
- Markl, Voronov
- 2003
(Show Context)
Citation Context ... P is 2 the restriction of the connected composition product Q ⊠c P on graphs of G 1 2 2 . 1 corresponds to the notion of 2-PROP defined by M. A monoid for the product ✷ 1 2 Markl and A.A. Voronov in =-=[MV]-=- and introduced by M. Kontsevich. Once again, one can apply Theorem 9. The free 1 2-PROP on an S-bimodule V is given the sum on graphs with one vertex in the middle, grafted above by trees (without le... |

13 | On the invertibility of quantization functors
- Enriques, Etingof
(Show Context)
Citation Context ...e refer the reader to [V]. From the previous theorem, one can get the description of the free prop on an S-bimodule V . We find the same construction of the free prop as B. Enriquez and P. Etingof in =-=[EE]-=- in terms of forests of graphs without levels. Recall that we have the following inclusions of monoidal abelian categories (see [V] Section 1) (V ect, ⊗k, k) ↩→ (S-Mod, ◦, I) ↩→ (S-biMod, ⊠c, I), wher... |

13 |
La renaissance des opérades, Séminaire Bourbaki 1994/95, Astérisque
- Loday
- 1996
(Show Context)
Citation Context ...ut to be a monoid in an appropriate monoidal category. The best example is the notion of operad which is a monoid in the monoidal category of S-modules with the composition product ◦ (see J.-L. Loday =-=[L]-=- or J.P. May [M])). The example of the free properad is new. The other free monoids given here were already known but the construction of Section 3 gives a conceptual explanation for their particular ... |

12 |
Infinite loop spaces, Annals of Mathematics Studies 90
- Adams
- 1978
(Show Context)
Citation Context ...relation V ⊠c I ≃ I ⊠c V , which is equivalent to forget the levels. □ The notion of properad is a “connected” version of the notion of prop (see F.W. Lawvere [La], S. Mac Lane [MacL2] and J.F. Adams =-=[A]-=-). For more details about the link between these two notions we refer the reader to [V]. From the previous theorem, one can get the description of the free prop on an S-bimodule V . We find the same c... |

12 |
A Koszul duality for props, Trans
- Vallette
(Show Context)
Citation Context ...f props. For instance, Frobenius bialgebras and Lie bialgebras can be modelled by dioperads (see [G]) whereas associative bialgebras and involutive Lie bialgebras require the notion of properads (see =-=[V]-=-). These two notions are monoids which generate bigger props but such that the associated categories of models are the same. The (co)homology theories and the lax notion “up to homotopy” of a particul... |

7 |
De operads, algebras and modules. Operads: Proceedings of Renaissance Conferences
- May
- 1997
(Show Context)
Citation Context ...d in an appropriate monoidal category. The best example is the notion of operad which is a monoid in the monoidal category of S-modules with the composition product ◦ (see J.-L. Loday [L] or J.P. May =-=[M]-=-)). The example of the free properad is new. The other free monoids given here were already known but the construction of Section 3 gives a conceptual explanation for their particular form. 5.1. Free ... |

5 |
Cohomology of monoids in monoidal categories. Operads
- BAUES, JIBLADZE, et al.
- 1995
(Show Context)
Citation Context ...ct preserves coproducts on one side, G.M. Kelly gave a construction of the free monoid by means of a particular colimit in Equation (23.2) page 69 of [K] (see also H.J. Baues, M. Jibladze, A.Tonks in =-=[BJT]-=- Appendix B and C. Rezk [R] Appendix A). This construction applies to operads. Since the other monoidal products considered here do not preserve coproducts neither general colimits, we need to refine ... |

5 |
PROPped up graph cohomology, Preprint arXiv: math.QA/0307081
- Markl, Voronov
- 2003
(Show Context)
Citation Context ...This product is associative and has I for unit. Therefore, (S-biMod, ✷ 1 , I) is a 2 monoidal abelian category. A monoid in this category is a 1 2-prop, notion defined by M. Markl and A.A. Voronov in =-=[MV]-=- and introduced by M. Kontsevich [Ko, Ma]. Once again, we can apply Theorem 10. The free 1 2-prop on an S-bimodule V is given the sum on graphs with one vertex in the middle, grafted above by trees (w... |

5 | The Biderivative and A∞-bialgebras
- Saneblidze, Umble
(Show Context)
Citation Context ...inition 19). We can apply Theorem 10 in this case which gives the free special prop. Notice that this notion of special props corresponds to the notion of matrons defined by S. Saneblidze R. Umble in =-=[SU]-=- and is related to the notion of 2 3-prop of B. Shoikhet [Sh]. 5.5. Free colored operad. Roughly speaking, a colored operad is an operad where the operations have colors indexing the leaves and the ro... |

4 |
Free monoids
- Dubuc
- 1974
(Show Context)
Citation Context ...eral, the existence of the free monoid has been established, under some hypotheses, by M. Barr in [B]. When the monoidal product preserves colimits over the simplicial category, E. Dubuc described in =-=[D]-=- a construction for the free monoid. A general categorical answer was given by G.M. Kelly in [K] when the monoidal product preserves colimit on one side. Once again, its construction requires the tens... |

3 | Deformation theory of representations of prop(erad)s. arXiv:0707.0889 - Merkulov, Vallette |

2 |
On the invertibility of quantization functors, preprint arXiv:math.QA/0306212
- Enriquez, Etingof
- 2003
(Show Context)
Citation Context ... the reader to [V]. From the previous theorem, one can get the description of the free prop on an S-bimodule V . Hence, we find the same construction of the free prop as B. Enriquez and P. Etingof in =-=[EE]-=- in terms of forests of graphs without levels. One has the following inclusions of monoidal abelian categories (V ect, ⊗k, k) ֒→ (S-Mod, ◦, I) ֒→ (S-biMod, ⊠c, I), where the product ◦ of S-modules cor... |

1 |
Operads and PROPs, math.AT/0601129 to appear
- Markl
(Show Context)
Citation Context ...udy the deformation theory of these (bi)algebras, it is enough to prove Koszul duality theory for the simpler monoid. (For more details on these notions, we refer the reader to the survey of M. Markl =-=[Ma2]-=-). Denote by G 1 2 2 the set of 2-level connected graphs such that every vertices of the first level has only one output or such that every vertices of the second level has only one input (see Figure ... |

1 |
A concept of 2 -PROP and deformation theory of (co)associative bialgebras
- Shoikhet
(Show Context)
Citation Context ... the free special prop. Notice that this notion of special props corresponds to the notion of matrons defined by S. Saneblidze R. Umble in [SU] and is related to the notion of 2 3-prop of B. Shoikhet =-=[Sh]-=-. 5.5. Free colored operad. Roughly speaking, a colored operad is an operad where the operations have colors indexing the leaves and the root. The composition of such operations is null if the colors ... |

1 |
A Koszul duality for props, preprint, http://math.unice.fr/∼brunov/ publications/Koszuldualityforprops.ps Laboratoire J.A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice Cedex 02, France E-mail address : brunov@math.unice.fr URL : http://math.uni
- Vallette
(Show Context)
Citation Context ...ecall the definition of the monoidal category of S-bimodules with the connected composition product ⊠c. For a full treatment of S-bimodules and the related monoidal categories, we refer the reader to =-=[V]-=-. 8 □�� �� � � � � �� �� � � � � � �� �� � � �� � �� FREE MONOID IN MONOIDAL ABELIAN CATEGORIES Definition (S-bimodules). An S-bimodule P is a collection (P(m, n))m, n∈N of modules over the symmetric... |