## A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE OVER POSETS

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@MISC{Roberts_acohomological,

author = {John E. Roberts and Giuseppe Ruzzi},

title = {A COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE OVER POSETS},

year = {}

}

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### Abstract

Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1–cocycle as a principal bundle, a connection turns out to be a 1–cochain associated in a suitable way with this 1–cocycle; the curvature of a connection turns out to be its 2–coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem. 1.

### Citations

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The Geometry of Four-Manifolds
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(Show Context)
Citation Context ... that a connection u is flat if, and only if, u is a 1–cocycle. Then, as an immediate consequence of Proposition 3.11, we have a poset version of a classical result of the theory of principal bundles =-=[11, 7]-=-. 4.6. Corollary. There is, up to equivalence, a 1-1 correspondence between flat connections of K with values in G and group homomorphisms from π1(K) into G. The existence of nonflat connections will ... |

472 |
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Citation Context ... that a connection u is flat if, and only if, u is a 1–cocycle. Then, as an immediate consequence of Proposition 3.11, we have a poset version of a classical result of the theory of principal bundles =-=[11, 7]-=-. 4.6. Corollary. There is, up to equivalence, a 1-1 correspondence between flat connections of K with values in G and group homomorphisms from π1(K) into G. The existence of nonflat connections will ... |

261 |
Simplicial objects in algebraic topology
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Citation Context ...ry structure of Σ∗(K) will be introduced and analyzed in the third subsection. Throughout this section, we shall consider a poset K and denote its order relation by ≤. References for this section are =-=[14, 20, 22]-=-. 2.1. Symmetric simplicial sets. A simplicial set Σ∗ is a graded set indexed by the non-negative integers equipped with maps ∂i : Σn → Σn−1 and σi : Σn → Σn+1, with 0 ≤ i ≤ n, satisfying the followin... |

122 |
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Citation Context ...entity ddv = ι. One motivation for studying the cohomology of a poset K with values in a non-Abelian group comes from the algebraic approach to quantum field theory. The leading idea of this approach =-=[9]-=- is that all the physical content of a quantum system is encoded in the observable net, an inclusion preserving correspondence which associates to any open ands864 JOHN E. ROBERTS AND GIUSEPPE RUZZI b... |

96 |
The algebra of oriented simplexes
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Citation Context ...les. A. Some results on n–categories In this appendix we explain the definition of the category I(C), and how to derive the categories nG and the form of a 3–cochain. References for this appendix are =-=[16, 23, 2, 19]-=-. The category I(C). Consider an n–category C and let ⋄ be a composition law of C. Given an arrow t, the left and the right ⋄–units of t are the arrows l⋄(t) r⋄(t) of C satisfying the relations l⋄(t) ... |

47 | Generalized measures in gauge theory
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Citation Context ...n G by U 1 (K, G). This definition of a connection is related to the notion of the link operator in a lattice gauge theory ([6]) and to the notion of a generalized connection in loop quantum gravity (=-=[1, 13]-=-). Both the link operator and the generalized connection can be seen as a mapping A which associates an element A(e) of a group G to any oriented edge e of a graph α, and enjoys the following properti... |

43 | The classifying space of a crossed complex
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Citation Context ...⋆ = ⋄. 3.2. Non-Abelian cohomology. Our approach to the non-Abelian cohomology of a poset will be based on n–categories. But it is worth pointing out that crossed complexes could be used instead (see =-=[5]-=-). In this approach cocycles turn out to be morphisms from a crossed complex associated with Σ∗(K) to a suitable target crossed complex. This approach might be convenient for studying higher homotopy ... |

41 | Higher Gauge Theory
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Citation Context ...quations (see below).sA COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE 865 That an n–category is the right set of coefficients for a non-Abelian cohomology can be understood by borrowing from =-=[3]-=- the following observation. Assume that × is Abelian, that is, f × g equals g × f whenever the compositions are defined. Assume that ⋄–units are ×–units. Let 1, 1 ′ be, respectively, a left and a righ... |

40 |
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Citation Context ...the notion of the first homotopy group for a symmetric simplicial set associated with a poset. A purely combinatorial definition of homotopy groups for arbitrary simplicial sets has been given by Kan =-=[10]-=- (see also [14]). This definition is very easy to handle in the case of simplicial sets satisfying the extension condition. Recall that a simplicial set Σ∗ satisfies the extension condition if given 0... |

39 |
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Citation Context ...ry structure of Σ∗(K) will be introduced and analyzed in the third subsection. Throughout this section, we shall consider a poset K and denote its order relation by ≤. References for this section are =-=[14, 20, 22]-=-. 2.1. Symmetric simplicial sets. A simplicial set Σ∗ is a graded set indexed by the non-negative integers equipped with maps ∂i : Σn → Σn−1 and σi : Σn → Σn+1, with 0 ≤ i ≤ n, satisfying the followin... |

30 | Differential geometry of gerbes
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(Show Context)
Citation Context ...inting out some analogies between the theory of connections, as presented in this paper, and that developed in synthetic geometry by A. Kock [12], and in algebraic geometry by L. Breen and W. Messing =-=[4]-=-. The contact point with our approach resides in the fact that both of the other approaches make use of a combinatorial notion of differential forms taking values in a group G. So in both cases connec... |

24 |
Local cohomology and superselection structure
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(Show Context)
Citation Context ... studying charged sectors of the observable net: the charge transporters of sharply localized charges are 1–cocycles of the poset taking values in the group of unitary operators of the observable net =-=[15]-=-. The attempt to include more general charges in the framework of algebraic quantum field theory, charges of electromagnetic type in particular, has led one to derive higher cocycles equations, up to ... |

14 |
Topological Measure and Graph-Differential Geometry on the Quotient Space of Connections Int
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Citation Context ...n G by U 1 (K, G). This definition of a connection is related to the notion of the link operator in a lattice gauge theory ([6]) and to the notion of a generalized connection in loop quantum gravity (=-=[1, 13]-=-). Both the link operator and the generalized connection can be seen as a mapping A which associates an element A(e) of a group G to any oriented edge e of a graph α, and enjoys the following properti... |

12 | Combinatorics of curvature, and the Bianchi identity
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Citation Context ...tricted to inflating simplices. 4.8. Remark. It is worth pointing out some analogies between the theory of connections, as presented in this paper, and that developed in synthetic geometry by A. Kock =-=[12]-=-, and in algebraic geometry by L. Breen and W. Messing [4]. The contact point with our approach resides in the fact that both of the other approaches make use of a combinatorial notion of differential... |

8 |
Mathematical aspects of local cohomology. Algbres d’oprateurs et leurs applications en physique mathmatique (Proc
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Citation Context ...include more general charges in the framework of algebraic quantum field theory, charges of electromagnetic type in particular, has led one to derive higher cocycles equations, up to the third degree =-=[16, 17]-=-. The difference, with respect to the Abelian case, is that a n– cocycle equation needs n composition laws. Thus in non-Abelian cohomology instead, for example, of trying to take coefficients in a non... |

8 |
A Survey of Local Cohomology
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Citation Context ...eories to the exigencies of algebraic quantum field theory. If successful, this should allow one to uncover structural features of gauge theories. Some earlier ideas in this direction may be found in =-=[17]-=-. In mathematics, a gauge theory may be understood as a principal bundle over a manifold together with its associated vector bundles. For applications to physics, the manifold in question is spacetime... |

6 |
An introduction to n-categories, Category theory and computer science (Santa Margherita Ligure
- Baez
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Citation Context ...les. A. Some results on n–categories In this appendix we explain the definition of the category I(C), and how to derive the categories nG and the form of a 3–cochain. References for this appendix are =-=[16, 23, 2, 19]-=-. The category I(C). Consider an n–category C and let ⋄ be a composition law of C. Given an arrow t, the left and the right ⋄–units of t are the arrows l⋄(t) r⋄(t) of C satisfying the relations l⋄(t) ... |

4 | Finite sets and symmetric simplicial sets
- Grandis
(Show Context)
Citation Context ...AL DESCRIPTION OF CONNECTIONS AND CURVATURE 857 respectively; 0–simplices by the letter a; 1–simplices by b; 2–simplices by c and generic n–simplices by x. A simplicial set Σ∗ is said to be symmetric =-=[8]-=- if there are mappings τi : Σn(K) → Σn(K) for n ≥ 1 and i ∈ {0, . . . n − 1}, satisfying the relations τi τi = 1, τi τi−1τi = τi−1 τiτi−1, τj τi = τi τj i < j − 1; ∂jτi = τi−1 ∂j i > j, ∂i+1 = ∂iτi, ∂... |

4 |
Homotopy of posets, net-cohomology, and theory of superselection sectors in globally hyperbolic spacetimes
- Ruzzi
(Show Context)
Citation Context ...ry structure of Σ∗(K) will be introduced and analyzed in the third subsection. Throughout this section, we shall consider a poset K and denote its order relation by ≤. References for this section are =-=[14, 20, 22]-=-. 2.1. Symmetric simplicial sets. A simplicial set Σ∗ is a graded set indexed by the non-negative integers equipped with maps ∂i : Σn → Σn−1 and σi : Σn → Σn+1, with 0 ≤ i ≤ n, satisfying the followin... |

2 |
Quarks, gluons and lattices. Cambridges UP
- Creutz
- 1983
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Citation Context ...∂1c), for any c ∈ Σ inf 2 (K). We denote the set of connection with values in G by U 1 (K, G). This definition of a connection is related to the notion of the link operator in a lattice gauge theory (=-=[6]-=-) and to the notion of a generalized connection in loop quantum gravity ([1, 13]). Both the link operator and the generalized connection can be seen as a mapping A which associates an element A(e) of ... |

2 | Net cohomology and its applications to quantum field theory - Roberts - 1980 |

2 |
The cohomology and homology of quantum field theory In : Quantum fields and quantum space time Cargése
- Roberts
- 1996
(Show Context)
Citation Context ...e present paper. Our first aim is to introduce an n–category associated with a group G to be used as set of coefficients for the cohomology of the poset K. To this end, we draw on a general procedure =-=[19]-=- associating to an n–category C, satisfying suitable conditions, an (n+1)–category I(C). This construction allows one to define the (n+1)–coboundary of a n–cochain in C as an (n + 1)–cochain in I(C), ... |

2 |
Local cohomology and superselection
- Roberts
- 1977
(Show Context)
Citation Context ... studying charged sectors of the observable net: the charge transporters of sharply localized charges are 1-cocycles of the poset taking values in the group of unitary operators of the observable net =-=[15]-=-. The attempt to include more general charges in the framework of algebraic quantum field theory, charges of electromagnetic type in particular, has led one to derive higher cocycles equations, up to ... |

1 |
Principal bundles over posets
- Roberts, Ruzzi, et al.
(Show Context)
Citation Context ...understand the family of functions {za0,a1}, defined by (45), as a 1–cocycle of a poset, in the sense of the Čech cohomology, with respect to the minimal covering of the poset. In a forthcoming paper =-=[21]-=-, we shall see that such functions are nothing but the transition functions of a principal bundle over a poset. We shall also see that such bundles can be mapped in to locally constant bundles over M ... |

1 |
Higher gauge theory. math.DG/0511710 [4
- Baez, Schreiber
- 1991
(Show Context)
Citation Context ...quations (see below).sA COHOMOLOGICAL DESCRIPTION OF CONNECTIONS AND CURVATURE 865 That an n-category is the right set of coefficients for a non-Abelian cohomology can be understood by borrowing from =-=[3]-=- the following observation. Assume that * is Abelian, that is, f * g equals g * f whenever the compositions are defined. Assume that \Pi -units are *-units. Let 1, 10 be, respectively, a left and a ri... |

1 |
The geometry of four-manifolds
- Quarks, UP, et al.
- 1983
(Show Context)
Citation Context ...c), for any c 2 \Sigma inf2 (K). We denote the set of connection with values in G by U1(K, G). This definition of a connection is related to the notion of the link operator in a lattice gauge theory (=-=[6]-=-) and to the notion of a generalized connection in loop quantum gravity ([1, 13]). Both the link operator and the generalized connection can be seen as a mapping A which associates an element A(e) of ... |