Affine parts of monads

Lindner, Harald

doi:10.1007/bf01222782

Cited by 9 publications

(11 citation statements)

References 3 publications

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“…This section first recalls the basic theory of affine monads -which preserve the final object 1 -following [51,55,25]. It then digs deeper into affineness and introduces a slightly stronger notion, called 'strong affiness', following [33].…”

Section: Affineness and Strong Affiness Of Monadsmentioning

confidence: 99%

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From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These 'probabilistic' models are called commutative effectuses, and are the focus of attention here. The paper describes the main known 'probability' monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate-and state-transformers, normalisation and conditioning of states.

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Section: Affineness and Strong Affiness Of Monadsmentioning

confidence: 99%

“…It is known for a long time that the 'affine part' of a monad can be extracted via pullbacks, see [55] (or also [25]). Here we shall relate this affine part to 'causal' maps in Kleisli categories of monads.…”

Section: Lemma 10 Let T Be An Affine Commutative Monad On a Distributmentioning

confidence: 99%

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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“…As a first example we will recall the strong probability monad of Giry, see [Law62,Gir82]. Lin79,Jac16], if C has a terminal object 1 and the unique morphism:…”

Section: Monadsmentioning

confidence: 99%

Quasi-Measurable Spaces

Forré¹

2021

Preprint

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We introduce the categories of quasi-measurable spaces, which are slight generalizations of the category of quasi-Borel spaces, where we now allow for general sample spaces and less restrictive random variables, quasi-measurable spaces and maps. We show that each category of quasi-measurable spaces is bi-complete and cartesian closed. We also introduce several different strong probability monads. Together these constructions provide convenient categories for higher probability theory that also support semantics of higherorder probabilistic programming languages in the same way as the category of quasi-Borel spaces does.An important special case is the category of quasi-universal spaces, where the sample space is the set of the real numbers R together with the σ-algebra of all universally measurable subsets. The induced σ-algebras on those quasiuniversal spaces then have explicit descriptions in terms of intersections of Lebesgue-complete σ-algebras. A central role is then played by countably separated quasi-universal spaces and universal quasi-universal spaces, which replace the role of standard Borel measurable spaces. We then go on and prove in this setting, besides others, a Fubini theorem, a disintegration theorem for Markov kernels, a Kolmogorov extension theorem and a conditional de Finetti theorem. We also translate our findings into properties of the corresponding Markov category of Markov kernels between universal quasiuniversal spaces.Furthermore, we formalize probabilistic graphical models like causal Bayesian networks in terms of quasi-universal spaces and prove a global Markov property for them. For this we need to translate the notion of transitional conditional independence into this setting and study its (asymmetric) separoid

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“…In this section we recall what it means for a monad to be affine (see [17,18,10]), and introduce a slightly stronger notion. We describe basic properties and examples.…”

Section: Affine and Strongly Affine Monadsmentioning

confidence: 99%

“…It is known for a long time that the 'affine part' of a monad can be extracted via pullbacks, see [18] (or also [10]). Here we shall relate this affine part to 'causal' maps in Kleisli categories of monads.…”

Section: Affine Parts Of Monads and Causal Mapsmentioning

confidence: 99%

Affine Monads and Side-Effect-Freeness

Jacobs

2016

Coalgebraic Methods in Computer Science

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Abstract. The notions of side-effect-freeness and commutativity are typical for probabilistic models, as subclass of quantum models. This paper connects these notions to properties in the theory of monads. A new property of a monad ('strongly affine') is introduced. It is shown that for such strongly affine monads predicates are in bijective correspondence with side-effect-free instruments. Also it is shown that these instruments are commutative, in a suitable sense, for monads which are commutative (monoidal).

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Affine parts of monads

Cited by 9 publications

References 3 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

Quasi-Measurable Spaces

Affine Monads and Side-Effect-Freeness

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