## Generalized operads and their inner cohomomorphisms, arXiv:math.CT/ 0609748 (2006)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Borisov06generalizedoperads,

author = {Dennis V. Borisov and Yuri I. Manin},

title = {Generalized operads and their inner cohomomorphisms, arXiv:math.CT/ 0609748},

year = {2006}

}

### OpenURL

### Abstract

Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =

### Citations

363 | Gromov-Witten classes, quantum cohomology and enumerative geometry
- Kontsevich, Manin
- 1994
(Show Context)
Citation Context ...n which components of the respective operad are supposed to lie. Operad itself for us is a functor from a category of labeled graphs to another symmetric monoidal category, as was stressed already in =-=[KoMa]-=-, [GeKa2] and many other works. We decided to spell out the underlying formalism in the Sec. 1 of this paper. If we appear to be too fussy e.g. in the Definition 1.3, this is because we found out that... |

327 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...nt functor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] [Bl], [CaGa], [Cra], =-=[Q]-=-, [R], [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful functor are quite mild. M... |

230 |
Koszul duality for operads
- Ginzburg, Kapranov
- 1994
(Show Context)
Citation Context ...roduct, although, as we have seen above, their existence is the most persistent phenomenon, even when the neat package (◦, •, !) cannot be preserved. a) Binary quadratic operads. The pioneering paper =-=[GiKa]-=- defined (◦, •, !) for binary quadratic (ordinary) operads. Their construction uses a description of operads as monoids in (COLL, ⊠), a category of collections endowed with a monoidal structure ⊠ (in ... |

133 |
Review of the elements of 2-categories
- Kelly, Street
(Show Context)
Citation Context ...neralized triple that under some conditions can be transformed into a usual triple. In such cases we will say that the triple is operad–like. The technique of lax morphisms was invented long ago (cf. =-=[KS]-=-) and applied recently to the case of operads in [Bat]. We proceed as follows: we start with the well known notion of strict pseudo– operads in categories, we organize them in a category and then exte... |

101 |
Topics in noncommutative geometry
- Manin
- 1991
(Show Context)
Citation Context ... F ⊗ B1 (all tensor products being taken over k). A morphism (F, u) → (F ′ , u ′ ) in A ⇒ B is a homomorphism of algebras v : F → F ′ such that u ′ = (v ⊗ idB) ◦ u. The following result was proved in =-=[Ma3]-=- (see. Prop. 2.3 in Chapter 4): 0.1.1. Theorem. The category A ⇒ B has an initial object (E, δ : A → E ⊗ B) 1 2000 Mathematics Subject Classification: 18D50, 18D10, 20C30. Keywords and phrases: Operad... |

96 |
Handbook of Categorical Algebra 1: Basic Category Theory
- Borceux
- 1994
(Show Context)
Citation Context ...that A has all coequalizers and for every A ∈ A the functor U(A)⊗− : C → C has a left adjoint. Then the functor A○− : A → A has a left adjoint as well. Proof. It is clear that U is a monadic functor (=-=[Bo1]-=-, Definition 4.4.1) and by our construction U(A○−) = U(A)⊗U(−), therefore (e.g. [Bo2], Theorem 4.5.6) since U(A)⊗− has a left adjoint, so does A○−. This shows our assertion. Here we define cohom(A, B)... |

95 |
Stacks of stable maps and Gromov-Witten invariants
- Manin
- 1996
(Show Context)
Citation Context ...Getzler for his remarks relating to model categories and the theory of operads in 2-categorical setting. §1. Background 1.1. Graphs. We define objects of the category of (finite) graphs as in [KoMa], =-=[BeMa]-=-, [GeKa2]. Geometric realizations of our graphs are not necessarily connected. This allows us to introduce a monoidal structure “disjoint union” on graphs (cf. 1.2.4), and to consider certain morphism... |

81 | Homological algebra of homotopy algebras - Hinich - 1997 |

70 | Modular operads
- Getzler, Kapranov
- 1998
(Show Context)
Citation Context ...components of the respective operad are supposed to lie. Operad itself for us is a functor from a category of labeled graphs to another symmetric monoidal category, as was stressed already in [KoMa], =-=[GeKa2]-=- and many other works. We decided to spell out the underlying formalism in the Sec. 1 of this paper. If we appear to be too fussy e.g. in the Definition 1.3, this is because we found out that uncritic... |

69 |
Handbook of Categorical Algebra 2, Categories and Structures
- Borceux
- 1994
(Show Context)
Citation Context ...” with empty set as the unit object. It exists, but is neither unique, nor completely obvious: what is the “disjoint union of a set with itself”? One way to introduce such a structure is described in =-=[Bo2]-=-, Example 6.1.9. We will focus on a small category of finite sets of all cardinalities and sketch the following method which neatly accounts for proliferation of combinatorics of symmetric groups in t... |

62 | 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 ...l structure is not biadditive, which is the case of operads. The resulting weight grading of the free monoid can be used to define weight graded quotients, analogs of graded associative algebras: see =-=[Fr]-=-, [Va1], [Va2]. The subcategory of ordinary operads with presentation considered in [GiKa] consists of weight graded operads generated by their binary parts (values on corollas with two inputs), with ... |

60 |
Sheafifiable homotopy model categories
- Beke
(Show Context)
Citation Context ... L is a cofibrant replacement functor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. =-=[Bek]-=- [Bl], [CaGa], [Cra], [Q], [R], [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful ... |

50 |
Tannakian categories. Hodge cycles, motives, and Shimura varieties
- Deligne, Milne
- 1982
(Show Context)
Citation Context ...of their respective flag and vertex sets, and ∂, j act on both parts as they used to. Finally, for any finite family {τs | s ∈ S}, we can define ∐ s τs functorially in {τs} and S as is spelled out in =-=[DeMi]-=-. 1.2.5. Atomization of a morphism. Let h : τ → σ be a morphism of graphs. We define its atomization as a commutative diagram of the following form: ∐ v∈Vσ τv ‘ hv ∐ �� v∈Vσ σv (1.3) k τ h � σ ◦σ Here... |

42 |
Toposes, Triples, and Theories. Grundlehren der mathematischen Wissenschaften
- Barr, Wells
- 1985
(Show Context)
Citation Context ...so on C. We can infer it easily from the adjoint lifting theorem, if we assume that A has all coequalizers. For the question of when a category of algebras over a triple has all coequalizers see e.g. =-=[BarW]-=-, Section 9.3. 3.1.4. Proposition. Suppose that A has all coequalizers and for every A ∈ A the functor U(A)⊗− : C → C has a left adjoint. Then the functor A○− : A → A has a left adjoint as well. Proof... |

41 |
Some remarks on Koszul algebras and quantum groups Annales de l’institut Fourier
- Manin
- 1987
(Show Context)
Citation Context ...as the canonical structure of a bialgebra. 0.2. Interpretation and motivation. Theorem 0.1.1 was the base of the approach to quantum groups as symmetry objects in noncommutative geometry discussed in =-=[Ma1]-=-–[Ma4]. Namely, consider PAlg as a category of function algebras on “quantum linear spaces” so that the category of quantum linear spaces themselves will be PAlg op . Then cohomomorphism algebras corr... |

33 |
An n-categorical pasting theorem
- Power
- 1991
(Show Context)
Citation Context ...∗ is composition of natural transformations, as defined in 4.1.1. }) : }), where It is easy to see that pseudo-operads in categories and lax morphisms form a category. From the the pasting theorem of =-=[Pow]-=-, we know that composition of ζ’s is associative and therefore composition of the whole morphisms is associative. There is an identity lax morphism for every category, given by the identity functor an... |

32 | The Eckmann-Hilton argument, higher operads and En-spaces
- Batanin
(Show Context)
Citation Context ...ansformed into a usual triple. In such cases we will say that the triple is operad–like. The technique of lax morphisms was invented long ago (cf. [KS]) and applied recently to the case of operads in =-=[Bat]-=-. We proceed as follows: we start with the well known notion of strict pseudo– operads in categories, we organize them in a category and then extend it to include lax morphism of operads, which satisf... |

30 | Homogeneous Algebras
- Berger, Dubois-Violette, et al.
- 2002
(Show Context)
Citation Context ...perads (and cooperads). Notice that the cohomology spaces of M0,n+1, components of the Quantum Cohomology cooperad, are quadratic algebras (Keel’s theorem). b) N–homogeneous algebras. It was shown in =-=[BerDW]-=- that similar results hold for the category HNalg of homogeneous algebras generated in degree 1 with relations generated in degree N, for any fixed N ≥ 2. If one continues to denote31 by RA ⊂ A ⊗N 1 ... |

30 |
Quantum groups and non-commutative geometry,” preprent Montreal University CRM-1561
- Manin
- 1988
(Show Context)
Citation Context ...”, and coendomorphism algeras, after passing to Hopf envelopes, become Hopf algebras of symmetries. (In fact, to obtain the conventional quantum3 groups, one has to add some “missing relations”, cf. =-=[Ma2]-=-, which also can be done functorially). In this paper we present several layers of generalizations of Theorem 0.1.1. The first step consists in extending it to operads with presentation and algebras o... |

28 | Koszul duality for dioperads - Gan |

27 | Cyclic operads and cyclic homology - Getzler, Kapranov - 1995 |

27 |
Spaces of algebra structures and cohomology of operads
- Rezk
- 1996
(Show Context)
Citation Context ...nctor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] [Bl], [CaGa], [Cra], [Q], =-=[R]-=-, [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful functor are quite mild. Moreov... |

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

25 | Koszul and Gorenstein properties for homogeneous algebras
- Berger, Marconnet
- 2003
(Show Context)
Citation Context ... Formulas (2.22)–(2.24) still hold in the new setup. Unit object for • is now k[ε]/(ε N ). As a consequence, Koszul complexes become N–complexes leading to an interesting new homological effects: see =-=[BerM]-=- and references therein. c) Homogeneous algebras generated in degree one. This case is treated in Chapter 3 of [PP] and in Section 1.3 of [GrM]; the approaches in these two papers nicely complement ea... |

15 | Prop profile of deformation quantization and graph complexes with loops and wheels, arXiv:math/0412257v7 - Merkulov - 2007 |

14 | Quillen closed model structures for sheaves
- Crans
- 1995
(Show Context)
Citation Context ...placement functor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] [Bl], [CaGa], =-=[Cra]-=-, [Q], [R], [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful functor are quite mi... |

14 | Simulating Quantum Computation by Contracting Tensor Networks,” quant-ph/0511069
- Markov, Shi
- 2005
(Show Context)
Citation Context ...dy knot invariants, Feynman perturbation series etc. c) Operads and their algebras are a formalization of computational processes and devices, in particular, tensor networks and quantum curcuits, cf. =-=[MarkSh]-=-, [Zo] and references therein. With this in mind, we describe general endomorphism operads in 2.5 below. It is interesting to notice that the classical theory of recursive functions must refer to a ve... |

13 | New model categories from old
- Blanc
- 1996
(Show Context)
Citation Context ...a cofibrant replacement functor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] =-=[Bl]-=-, [CaGa], [Cra], [Q], [R], [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful funct... |

13 |
Quadratic algebras, University Lecture Series, 37
- Polishchuk, Positselski
- 2005
(Show Context)
Citation Context ...omplexes become N–complexes leading to an interesting new homological effects: see [BerM] and references therein. c) Homogeneous algebras generated in degree one. This case is treated in Chapter 3 of =-=[PP]-=- and in Section 1.3 of [GrM]; the approaches in these two papers nicely complement each other. White product (2.21) and its HNalg–version extend to this larger category as (A ◦ B)n := An ⊗ Bn (Segre p... |

11 |
Free monoid in monoidal abelian categories, arXiv : math.CT/0411543
- Vallette
- 2004
(Show Context)
Citation Context ...This description was rather neglected here (see Appendix), but it makes clear the analogy between the tensor algebra of a linear space and the free operad generated by a collection V . B. Vallette in =-=[Va3]-=- describes a construction of a free monoid which allows him to treat the cases when the relevant monoidal structure is not biadditive, which is the case of operads. The resulting weight grading of the... |

9 |
Closed model structures for algebraic models of n-types
- Cabello, Garzn
- 1995
(Show Context)
Citation Context ...brant replacement functor on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] [Bl], =-=[CaGa]-=-, [Cra], [Q], [R], [S]. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful functor are q... |

9 | Manin products, Koszul duality, Loday algebras and Deligne conjecture - Vallette |

8 |
Multiple zeta-motives and moduli spaces M0,n
- Manin
(Show Context)
Citation Context ...o two different morphisms in Γ, with two different targets. Combinatorics of unoriented cyclic labeled trees is very essential in the description of the topological operad of real points M0,∗(R), cf. =-=[GoMa]-=-. 1.4. Ground categories G. Operads of various types in this paper will be defined as certain functors from a category of labeled graphs Γ to a symmetric monoidal category (G, ⊗) which will be called ... |

8 | Operads and PROPs
- Markl
- 2008
(Show Context)
Citation Context ...2). 0.3. The phantom of the operad. The next extension of Theorem 0.1.1 involves replacing operads by any of the related structures a representative list of which the reader can find, for example, in =-=[Mar]-=-: May and Markl operads, cyclic operads, modular operads, PROPS, properads, dioperads etc. In this paper, we use for all of them the generic name “generalized operad”, or simply “operad”, and call ope... |

7 |
Extended modular operad
- Manin
- 2004
(Show Context)
Citation Context ...led colors). An I–colored graph is a graph τ together with a map Fτ → I such that two halves14 of each edge get the same color. Morphisms are restricted by the condition that h F preserves color. In =-=[LoMa]-=-, a topological operad was studied governed by a category of colored graphs with two–element I = {black, white}. Halves of an edge in this category are always white. e) Cyclic labeling. A cyclic label... |

5 | Twisted internal coHom objects in the category of quantum spaces
- Grillo, Montani
(Show Context)
Citation Context ...e explicit descriptions of cohomomorphism objects, whenever they are known, in particular, for quadratic and more general N–homogeneous algebras and operads. Notice that Theorem 0.1.1 was extended in =-=[GrM]-=- to include the case of twisted tensor products of algebras: see also [Ma4] where the latter appeared in the construction of the De Rham complex of quantum groups and spaces. For further developments ... |

4 |
Introduction to bicategories. 1967 Reports of the Midwest Category Seminar
- Benabou
(Show Context)
Citation Context ...ralized triples on C will be denoted by T(C). To justify the term “generalized triple” we give the following example, which is illustrative but inessential in our considerations. It was considered in =-=[Ben]-=- Section 5.4. 4.7.2. Example. Let P be the strict operad E. A lax representation of E on a category C is simply triple on C in the usual meaning of the term. Indeed such representation consists first ... |

4 |
Strong homotopy theory of cyclic sets
- Spaliński
- 1995
(Show Context)
Citation Context ... on A. It remains only to analyze when such a transport of model structure is possible. The general situation of a transport was considered in several papers e.g. [Bek] [Bl], [CaGa], [Cra], [Q], [R], =-=[S]-=-. For our purposes it is enough to consider locally presentable categories with cofibrantly generated model structures. In this case the conditions on the forgetful functor are quite mild. Moreover, w... |

3 | math.CO/0605256 Tensor networks and the enumeration of regular subgraphs
- Zograf
(Show Context)
Citation Context ...variants, Feynman perturbation series etc. c) Operads and their algebras are a formalization of computational processes and devices, in particular, tensor networks and quantum curcuits, cf. [MarkSh], =-=[Zo]-=- and references therein. With this in mind, we describe general endomorphism operads in 2.5 below. It is interesting to notice that the classical theory of recursive functions must refer to a very spe... |

2 | Resolution of colored operads and rectification of homotopy algebras - Berger, Moerdijk |

2 |
A Koszul duality for PROPs. Preprint math.AT/0411542
- Vallette
(Show Context)
Citation Context ... sometimes sensible to include in a category Γ of directed graphs only those, which have at least one input and least one output at each vertex (cf. the definition of reduced bimodules in Sec. 1.1 ov =-=[Va1]-=-). Another reason for excluding certain marginal (“unstable”) types of labeled corollas might be our desire to ensure essential finiteness of the categories denoted ⇒ σ in Sec. 1.5.5 (cf. a descriptio... |