## ∗-AUTONOMOUS CATEGORIES IN QUANTUM THEORY (2006)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Day06∗-autonomouscategories,

author = {Brian J. Day},

title = {∗-AUTONOMOUS CATEGORIES IN QUANTUM THEORY},

year = {2006}

}

### OpenURL

### Abstract

mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗-autonomous partially ordered sets A = (A, p, j, S)

### Citations

226 | Classical and quantum conformal field theory - Moore, Seiberg - 1989 |

97 |
On closed categories of functors
- Day
- 1970
(Show Context)
Citation Context ... the literature, an abstract definition of ∗-autonomous promonoidal structure being made in [3, §7]. Without going into much detail, we also note some features of the convolution [A , V ] (defined in =-=[1]-=-) of a given such A with a complete ∗-autonomous monoidal category V . A monoidal functor category of this type is a completion of A , with an appropriate universal property; it is always again ∗-auto... |

58 |
Effect algebras and unsharp quantum logics
- Foulis, Bennett
- 1994
(Show Context)
Citation Context ...c, a) = p(c, a, b), and these simultaneously take the value 1 iff the points a, b, and c are “collinear”. Example 6 (Generalized effect and difference algebras (cf. Kalmbach [5], Chapter 21; see also =-=[4]-=-)). Suppose the poset A has the structure of a (non-commutative, say) generalized effect algebra (A , ⊕, 0, ≤). Let { 1 iff a ⊕ b ≤ c p(a, b, c) = 0 else, { 1 iff a ≤ b A (a, b) = 0 else,4 BRIAN J. D... |

27 |
Submodular Functions and Electrical Networks
- Narayanan
- 1997
(Show Context)
Citation Context ...entity 0. In the following, each poset A is viewed as a category under and the base category is V = R ∞ ≥0 A (a, b) = { 1 iff a ≤ b 0 else, unless otherwise mentioned. Example 1 (Submodular functions =-=[7]-=-). Let E be a set and A = P(E) (discrete). For V = R∞ ≥0 , let { 1 iff (a ∪ b = c and a ∩ b = ∅) iff a + b = c p(a, b, c) = 0 else, Date: March 25, 2006. 12 BRIAN J. DAY and j(a) = { 1 iff a = ∅ 0 el... |

19 | Quantum categories, star autonomy, and quantum groupoids
- Day, Street
(Show Context)
Citation Context ...iv:math/0605037v1 [math.CT] 1 May 2006 BRIAN J. DAY 1. Introduction So-called ∗-autonomous, or “Frobenius”, category structures occur widely in mathematical quantum theory. This trend was observed in =-=[3]-=-, mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗-autonomous partially ordered sets A = (A , p, j, S) that appear i... |

19 | Frobenius monads and pseudomonoids
- Street
(Show Context)
Citation Context ...uction So-called ∗-autonomous, or “Frobenius”, category structures occur widely in mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in =-=[8]-=- with a general account of Frobenius monoids. Below we list some of the ∗-autonomous partially ordered sets A = (A , p, j, S) that appear in the literature, an abstract definition of ∗-autonomous prom... |

1 |
Locale geometry, Pacific
- Day
- 1979
(Show Context)
Citation Context ...(a, b, c) = 0 else, { 1 iff a is an identity j(a) = 0 else, and Sa = a −1 . Then p(a, b, Sc) = 1 iff abc = 1, so (A , p, j, S) is ∗-autonomous. Example 5 ((Non-commutative) probabalistic geometry (of =-=[2]-=-)). Let A be a poset with an associative promultiplication �� [0, 1] ⊂ R ∞ ≥0 , p : A op × A op × A where we interpret the value p(a, b, c) as the probability that the point c lies in the line through... |

1 |
Quantum measures and spaces (monograph
- Kalmbach
- 1998
(Show Context)
Citation Context ...iff while f is a lower convolution monoid iff f(a + b) ≥ f(a) + f(b) and f(0) ≥ 0, f(a + b) ≤ f(a) + f(b) and f(0) ≤ 0. Example 2. Let A = (A , ∨, ∧, 0, 1, S) be an orthomodular lattice (see Kalmbach =-=[5]-=- for example). Then the definitions { 1 iff a ∨ b = c and a⊥b p(a, b, c) = 0 else, { 1 iff a = b A (a, b) = 0 else, and j(a) = { 1 iff a = 0 0 else, yield a (discrete) ∗-autonomous promonoidal poset (... |