## Probabilities, Distribution Monads, and Convex Categories

Citations: | 1 - 1 self |

### BibTeX

@MISC{Jacobs_probabilities,distribution,

author = {Bart Jacobs},

title = {Probabilities, Distribution Monads, and Convex Categories},

year = {}

}

### OpenURL

### Abstract

Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]-probabilities. It will be shown that there are translations back-and-forth, in the form of an adjunction, between effect monoids and “convex ” monads. This convexity property is formalised, both for monads and for categories. In the end this leads to “triangles of adjunctions ” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1

### Citations

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(Show Context)
Citation Context ... 7 describe a close connection between fundamental mathematical structures in the context of probabilistic systems. 2 Preliminaries This paper assumes familiarity with basic category theory, see e.g. =-=[19,4,18,6]-=-. It uses coproducts/sums (+, 0), with coprojections κi : Xi → X1 + X2 and unique maps !: 0 → X. It also uses monoidal structure (⊗, I), with the standard (associativity and unit) isomorphisms. The no... |

208 | Toposes, Triples and Theories
- Barr, Wells
(Show Context)
Citation Context ... 7 describe a close connection between fundamental mathematical structures in the context of probabilistic systems. 2 Preliminaries This paper assumes familiarity with basic category theory, see e.g. =-=[19,4,18,6]-=-. It uses coproducts/sums (+, 0), with coprojections κi : Xi → X1 + X2 and unique maps !: 0 → X. It also uses monoidal structure (⊗, I), with the standard (associativity and unit) isomorphisms. The no... |

165 | A categorical semantics of quantum protocols
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(Show Context)
Citation Context ...ties are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval =-=[0, 1]-=-. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]-probabilities. It will be shown that there are translations back-and-forth, in... |

125 |
Categorical Logic and Type Theory
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- 1999
(Show Context)
Citation Context ... indexed category A op �� EA by X � �� A(X, 2). The fibres over X are by construction the same as maps X → 2 in the base category. Hence 2 ∈ A forms a generic object (or classifier of predicates, see =-=[13]-=-). Proof. Clearly, h ∗ (0) = 0 ◦ h = κ2 ◦ ! ◦ h = κ2 ◦ ! = 0, and similarly h ∗ (1) = 1. Now assume f ⊥ g for f, g : Y → 2, via a bound b: Y → 3. Then b ◦ h: X → 3 is trivially also a bound for f ◦ h ... |

79 |
Algebraic Theories
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(Show Context)
Citation Context ... 7 describe a close connection between fundamental mathematical structures in the context of probabilistic systems. 2 Preliminaries This paper assumes familiarity with basic category theory, see e.g. =-=[19,4,18,6]-=-. It uses coproducts/sums (+, 0), with coprojections κi : Xi → X1 + X2 and unique maps !: 0 → X. It also uses monoidal structure (⊗, I), with the standard (associativity and unit) isomorphisms. The no... |

70 |
Effect algebras and unsharp quantum logics
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(Show Context)
Citation Context ...(in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1 Introduction In the foundation of quantum mechanics so-called effect algebras =-=[9,8]-=- have emerged as mathematical structures that capture both probabilities and propositions in a single mathematical notion. The unit interval, with its partial addition operation, is a main example. It... |

69 |
Semantics of weakening and contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...monads on A. In the special case where A = Sets we write CnvFun = CnvFun(Sets) and CnvMnd = CnvMnd(Sets). A functor F satisfying the first requirement F (1) ∼ = 1 is sometimes called affine, see e.g. =-=[17,12]-=-. The notion of convex functor was introduced (and used) in [14]; here we focus on convex monads. The following observation is fundamental. Proposition 6.2 The Kleisli categories Kℓ(T ) and KℓN(T ) of... |

41 | A hierarchy of probabilistic system types
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- 2003
(Show Context)
Citation Context ...nd distribution monads, taking probabilities not from [0, 1] but from such an effect monoid. Distribution monads are frequently used in the abstract modeling of probabilistic state based systems (see =-=[5]-=- for an overview). This connection makes it possible to consider a wider range of systems, involving a more general notion of probability. Preprint submitted to Theoretical Computer Science 1 Septembe... |

38 | S.: New Trends in Quantum Structures - Dvurečenskij, Pulmannová - 2000 |

18 |
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- 1994
(Show Context)
Citation Context |

18 |
Bilinearity and cartesian closed monads
- Kock
- 1971
(Show Context)
Citation Context ...monads on A. In the special case where A = Sets we write CnvFun = CnvFun(Sets) and CnvMnd = CnvMnd(Sets). A functor F satisfying the first requirement F (1) ∼ = 1 is sometimes called affine, see e.g. =-=[17,12]-=-. The notion of convex functor was introduced (and used) in [14]; here we focus on convex monads. The following observation is fundamental. Proposition 6.2 The Kleisli categories Kℓ(T ) and KℓN(T ) of... |

9 |
Algebraic Approaches to Program Semantics. Texts and Monogr
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- 1986
(Show Context)
Citation Context ... id = [[κ1, κ1], κ2]: 2 + 1 = 3 → 2 = 1 + 1 sends 1, 2 ↦→ 1 and 3 ↦→ 2. This style of defining a partial operation via a bound is reminiscent of partially 9�� �� � � �� �� �� ��� additive categories =-=[3]-=-. In this homset A(X, 2) we further define: 1 def ( = X ! κ1 �� 1 ) �� 2 0 def ( = X ! κ2 �� 1 ) �� 2 f ⊥ ( def = X f �� 2 [κ2,κ1] ∼= ) �� 2 . Theorem 5.2 Let A be a convex category. (1) With the abov... |

9 |
The transition to unigroups
- Foulis, Greechie, et al.
- 1998
(Show Context)
Citation Context ... In fact, for each positive number M ∈ R the interval [0, M]R = {r ∈ R | 0 ≤ r ≤ M} is an example of an effect algebra, with r⊥ = M − r. More generally, so-called “interval effect algebras”, see e.g. =-=[10]-=- or [8, 1.4] can be obtained from ordered Abelian groups. This includes the “effects” on a Hilbert space H, consisting of the positive operators H → H below the identity. (3) A separate class of examp... |

9 |
Representation theorem for convex effect algebras
- Gudder, Pulmannová
- 1998
(Show Context)
Citation Context ...his requires that we reformulate the action via a tensor, like in (2). As a result, Hom(X, 2) becomes an “effect 14module” over Hom(1, 2). Such a module is the same as a “convex effect algebra”, see =-=[20]-=-. 6 Convex monads This section introduces convex monads and relates them to effect monoids via an adjunction. This extends the adjunction in [14] between convex functors and effect algebras. It is sim... |

8 | Quantum logic in dagger kernel categories. Order
- Heunen, Jacobs
- 2010
(Show Context)
Citation Context ..., the lattice of closed subsets of a Hilbert space is an orthomodular lattice and thus an effect algebra. This applies more generally to the kernel subobjects of an object in a dagger kernel category =-=[11]-=-. (4) Since Boolean algebras are (distributive) orthomodular lattices, they are also effect algebras. By distributivity, elements in a Boolean algebra are orthogonal if and only if they are disjoint, ... |

5 |
Scalars, monads and categories
- Coumans, Jacobs
- 2010
(Show Context)
Citation Context ...onoids. This may be seen as the main technical result of the paper. In the end the various connections will be organised in Section 7 in terms of “triangles of adjunctions”, following the paradigm of =-=[7]-=-. This paper builds on [14], where an adjunction between effect algebras and convex functors was established. The present paper extends this adjunction to effect monoids and convex monads, and adds th... |

3 | 2010): Convexity, duality, and effects
- Jacobs
(Show Context)
Citation Context ...s the main technical result of the paper. In the end the various connections will be organised in Section 7 in terms of “triangles of adjunctions”, following the paradigm of [7]. This paper builds on =-=[14]-=-, where an adjunction between effect algebras and convex functors was established. The present paper extends this adjunction to effect monoids and convex monads, and adds the notion of convex category... |

1 | Coreflections in algebraic quantum logic - Jacobs, Mandemaker - 1980 |