## ON DEFORMATIONS OF PASTING DIAGRAMS

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@MISC{Yetter_ondeformations,

author = {D. N. Yetter},

title = {ON DEFORMATIONS OF PASTING DIAGRAMS},

year = {}

}

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### Abstract

Abstract. We adapt the work of Power [14] to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov [10]. 1.

### Citations

248 |
On the deformation of rings and algebras
- Gerstenhaber
- 1964
(Show Context)
Citation Context ... D. N. YETTER 3. Deformations of categories, functors and natural transformations: definitions and elementary results The generalization of Gerstenhaber’s deformation theory from associative algebras =-=[7, 8]-=- to linear categories, or ’algebroids’ in the sense of Mitchell [13], is quite straight-forward, and both the one readily available source in which the construction has appeared [1] and unpublished le... |

96 |
The algebra of oriented simplexes
- Street
- 1987
(Show Context)
Citation Context ...ries what an ordinary diagram is to categories. A number of ways to formalize them have been developed. We will for the most part follow Power [14], whose approach mixing Street’s notion of computads =-=[15]-=- with a geometric adaptation of Johnson’s pasting schemes [11] avoids much of the combinatorial complexity of Johnson’s approach. Power’s description seems to be of the right generality for the presen... |

61 |
Rings with several objects
- Mitchell
- 1972
(Show Context)
Citation Context ...sformations: definitions and elementary results The generalization of Gerstenhaber’s deformation theory from associative algebras [7, 8] to linear categories, or ’algebroids’ in the sense of Mitchell =-=[13]-=-, is quite straight-forward, and both the one readily available source in which the construction has appeared [1] and unpublished lectures of Tsygan [16] treat it as a folk-theorem. The deformation th... |

57 | AA Voronov, Homotopy G-algebras and moduli space operad - Gerstenhaber - 1995 |

33 |
An n-categorical pasting theorem
- Power
- 1991
(Show Context)
Citation Context ...ng diagrams: definitions A pasting diagram is to n-categories what an ordinary diagram is to categories. A number of ways to formalize them have been developed. We will for the most part follow Power =-=[14]-=-, whose approach mixing Street’s notion of computads [15] with a geometric adaptation of Johnson’s pasting schemes [11] avoids much of the combinatorial complexity of Johnson’s approach. Power’s descr... |

16 |
On the deformation of algebra morphisms and diagrams
- Gerstenhaber, Schack
- 1983
(Show Context)
Citation Context ... commutative parts. We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack =-=[9]-=-, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov [10]. 1. Introduction It is the purpose of t... |

14 | The quantum tetrahedron in 3 and 4 dimensions
- Baez
- 1999
(Show Context)
Citation Context ...nstances of pasting diagrams. Consideration of not-necessarily abelian linear stacks is motivated by physical considerations in a prospective deformation quantization approach to quantum gravity (cf. =-=[2, 3, 4]-=-). It is also hoped that the present work may shed light, if only by analogy, on the difficulties arising in Elgueta’s deformation theory for monoidal bicategories [6]. Throughout we will consider all... |

13 | Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
- Johnson
- 1987
(Show Context)
Citation Context ...ys to formalize them have been developed. We will for the most part follow Power [14], whose approach mixing Street’s notion of computads [15] with a geometric adaptation of Johnson’s pasting schemes =-=[11]-=- avoids much of the combinatorial complexity of Johnson’s approach. Power’s description seems to be of the right generality for the present work, and we will deviate from it only to allow the descript... |

13 |
Functorial Knot Theory : Categories of Tangles
- Yetter
(Show Context)
Citation Context ...ansformations. The present work has a number of motivations. It initially grew out of a program to extend the author’s deformation theory for monoidal categories, functors and natural transformations =-=[5, 17, 18, 19]-=-, which deforms only the structure maps, to a theory in which the composition of the category, the arrow part of the monoidal product, and the structure maps are all deformed simultaneously. That prog... |

7 |
Deformations of (bi)tensor categories,” Cahier de Topologie et Géometrie Differentielle Catégorique (to appear
- Crane, Yetter
(Show Context)
Citation Context ...ansformations. The present work has a number of motivations. It initially grew out of a program to extend the author’s deformation theory for monoidal categories, functors and natural transformations =-=[5, 17, 18, 19]-=-, which deforms only the structure maps, to a theory in which the composition of the category, the arrow part of the monoidal product, and the structure maps are all deformed simultaneously. That prog... |

7 |
Cohomology and deformation theory of monoidal 2categories I
- Elgueta
(Show Context)
Citation Context ...ch to quantum gravity (cf. [2, 3, 4]). It is also hoped that the present work may shed light, if only by analogy, on the difficulties arising in Elgueta’s deformation theory for monoidal bicategories =-=[6]-=-. Throughout we will consider all categories to be small, if necessary by invoking the The author wishes to thank Kansas State University and the University of Pennsylvania for support of the sabbatic... |

7 | Hochschild cohomology and moduli spaces of strongly homotopy associative algebras
- Lazarev
(Show Context)
Citation Context ...logical analog: 4.3. Theorem. The normalized Hochschild complex ¯ C • (F, G) is a chain deformation retract of the Hochschild complex C • (F, G). Proof. The result follows from the same trick used in =-=[12]-=-. Call a cochain φ i-normalized if φ(f0, . . . , fn−1) is zero whenever fj is an identity arrow for any j ≤ i. The i-normalized cochains from a subcomplex C • i (F, G) of C • (F, G), and satisfy and C... |

6 | Braided deformations of monoidal categories and vassiliev invariants
- Yetter
- 1998
(Show Context)
Citation Context ...ansformations. The present work has a number of motivations. It initially grew out of a program to extend the author’s deformation theory for monoidal categories, functors and natural transformations =-=[5, 17, 18, 19]-=-, which deforms only the structure maps, to a theory in which the composition of the category, the arrow part of the monoidal product, and the structure maps are all deformed simultaneously. That prog... |

5 |
Categorical Geometry and the Mathematical Foundations of Quantum General Relativity; Contribution to the Cambridge University Press volume on Quantum Gravity
- Crane
(Show Context)
Citation Context ...nstances of pasting diagrams. Consideration of not-necessarily abelian linear stacks is motivated by physical considerations in a prospective deformation quantization approach to quantum gravity (cf. =-=[2, 3, 4]-=-). It is also hoped that the present work may shed light, if only by analogy, on the difficulties arising in Elgueta’s deformation theory for monoidal bicategories [6]. Throughout we will consider all... |

4 | What is the Mathematical Structure of Quantum Spacetime? arXiv:0706.4452
- Crane
- 2007
(Show Context)
Citation Context ...nstances of pasting diagrams. Consideration of not-necessarily abelian linear stacks is motivated by physical considerations in a prospective deformation quantization approach to quantum gravity (cf. =-=[2, 3, 4]-=-). It is also hoped that the present work may shed light, if only by analogy, on the difficulties arising in Elgueta’s deformation theory for monoidal bicategories [6]. Throughout we will consider all... |

4 |
The cohomology of an associative ring
- Gerstenhaber
- 1963
(Show Context)
Citation Context ... D. N. YETTER 3. Deformations of categories, functors and natural transformations: definitions and elementary results The generalization of Gerstenhaber’s deformation theory from associative algebras =-=[7, 8]-=- to linear categories, or ’algebroids’ in the sense of Mitchell [13], is quite straight-forward, and both the one readily available source in which the construction has appeared [1] and unpublished le... |

2 |
Moduli of linear categories: examples of higher stacks. lecture at IHES
- Anel
(Show Context)
Citation Context ...iative algebras [7, 8] to linear categories, or ’algebroids’ in the sense of Mitchell [13], is quite straight-forward, and both the one readily available source in which the construction has appeared =-=[1]-=- and unpublished lectures of Tsygan [16] treat it as a folk-theorem. The deformation theory for linear functors, or for that matter commutative diagrams of linear functors, is similarly a straight-for... |

2 |
Deformation quantization of gerbes. unpublished lecture at
- Tsygan
(Show Context)
Citation Context ...ries, or ’algebroids’ in the sense of Mitchell [13], is quite straight-forward, and both the one readily available source in which the construction has appeared [1] and unpublished lectures of Tsygan =-=[16]-=- treat it as a folk-theorem. The deformation theory for linear functors, or for that matter commutative diagrams of linear functors, is similarly a straight-forward generalization of work of Gerstenha... |

2 | Abelian categories of modules over a (lax) monoidal functor
- Yetter
(Show Context)
Citation Context |