## COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS

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@MISC{Kock_commutativemonads,

author = {Anders Kock},

title = {COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS},

year = {}

}

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### Abstract

Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.

### Citations

945 |
Categories for the working mathematician
- Lane
- 1971
(Show Context)
Citation Context ... Bill Lawvere for fruitful discussions in Nancy, and e-mail correspondence in the spring of 2011. 1. Monads, algebras, and linearity Recall the notion of monad T = (T, η, µ) on a category E, cf. e.g. =-=[16]-=- 6.1 and 6.2. Recall also the notion of T -algebra A = (A, α) for such monad; here, α : T (A) → A is the structure map for the given algebra. There is a notion of morphism of algebras (A, α) → (B, β),... |

747 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...in the present paper. The tensorial form has the advantage not of mentioning ⋔ at all, so makes sense even for endofunctors on monoidal categories that are not closed; this has been exploited e.g. in =-=[17]-=- and [3]. 3. Strong monads; partial linearity The composite of two strong endofunctors on the cartesian closed E carries a strength derived from the two given strengths. We give it explicitly for the ... |

145 |
Handbook of categorical algebra
- Borceux
- 1994
(Show Context)
Citation Context ...that the evaluation map is a map ev : (X ⋔ Y ) × X → Y. Since E is a monoidal category (with cartesian product as monoidal structure), one has the notion of when a category is enriched in E, cf. e.g. =-=[1]-=- II.6.2. And since this monoidal category E is in fact monoidal closed (with ⋔ as the closed structure), E is enriched in itself. For E-enriched categories, one has the notion of enriched functor, as ... |

49 | Strong functors and monoidal monads - Kock - 1972 |

49 |
Methodes mathematiques pour les sciences physiques, Enseignement des Sciences
- Schwartz
- 1961
(Show Context)
Citation Context ...a (continuous linear) functional on a space of functions. Schwartz argues that “the mathematical distributions constitute a correct mathematical definition of the distributions one meets in physics”, =-=[19]-=- p. 84. Our aim here is to present an alternative mathematical theory of distributions (of compact support), applicable also for the “tomato” example, but which does not depend on the “double dualizat... |

43 |
Monads on symmetric monoidal closed categories
- Kock
- 1970
(Show Context)
Citation Context ...D25, 46T30. Key words and phrases: monads, distributions, extensive quantities. c○ Anders Kock, 2012. Permission to copy for private use granted. 9798 ANDERS KOCK commutativity of T (in the sense of =-=[6]-=-). This is what makes for instance the notions of linearity/bilinearity work well. Generalities: We mainly compose maps from right to left (this is the default, and it is denoted g ◦ f); but occasiona... |

18 |
Bilinearity and cartesian closed monads
- Kock
- 1971
(Show Context)
Citation Context ...⋔ C. The T -algebra X ⋔ C thus constructed actually witnesses that the category E T of T - algebras is cotensored over E, cf. e.g. [1] II.6.5. The tensorial strength makes possible a description (cf. =-=[9]-=-) of the crucial notion of partial T -linearity (recall the use of the phrase T -linear as synonymous with T -algebra morphism). Let (B, β) and (C, γ) be T -algebras, and let X ∈ E be an arbitrary obj... |

15 | Closed categories generated by commutative monads
- Kock
- 1971
(Show Context)
Citation Context ...on “st”. We shall, however, need to consider two other equivalent manifestations of such “strength” structure on T , introduced in [6] (and proved equivalent to strength in [10], Theorem 1.3), and in =-=[8]-=-, called tensorial and cotensorial strength, respectively. To give a tensorial strength to the endofunctor T is to give, for any pair of objects X, Y in E a map X × T (Y ) t′′ X,Y ✲ T (X × Y ), satisf... |

10 |
Relative Functor Categories and Categories of Algebras
- Bunge
- 1969
(Show Context)
Citation Context ...S A THEORY OF DISTRIBUTIONS 107 or briefly, τ ◦ η = δ, (B being a fixed T -algebra here). In [13], we even denoted τX(x) by δx. 8. The object of T -linear maps The following construction goes back to =-=[2]-=-. Let (B, β) and (C, γ) be two T -algebras, with T a strong monad on E. We assume that E has equalizers. Then we can out of the object B ⋔ C carve a subobject B ⋔T C “consisting of” the T -linear maps... |

9 |
Synthetic Differential Geometry (London
- Kock
- 1981
(Show Context)
Citation Context ...ral in B ∈ E T and in X ∈ E, the latter naturality in the “extranatural” sense, [16] 9.4, which we shall recall. If we use elements to express equations (a well known and rigorous technique, cf. e.g. =-=[11]-=- II.1, even though objects in a category often are said to have no elements), the extraordinary naturality in X is expressed: for any f : Y → X, φ ∈ X ⋔ B and P ∈ T (Y ), 〈T (f)(P ), φ〉 = 〈P, φ ◦ f〉, ... |

8 |
Categories of space and of quantity
- Lawvere
- 1992
(Show Context)
Citation Context ...a canonical comparison to the distributions of functional analysis. Distributions of compact support form an important example of an extensive quantity, in the sense made precise by Lawvere, cf. e.g. =-=[14]-=-, and whose salient feature is the covariant functorial dependence on the space over which it is distributed. Thus, there is a covariant functor T , such that extensive quantities of a given type on a... |

4 | On double dualization monads - Kock - 1970 |

1 |
Categories, arXiv [math.RA
- Coumans, Jacobs, et al.
- 2010
(Show Context)
Citation Context ...esent paper. The tensorial form has the advantage not of mentioning ⋔ at all, so makes sense even for endofunctors on monoidal categories that are not closed; this has been exploited e.g. in [17] and =-=[3]-=-. 3. Strong monads; partial linearity The composite of two strong endofunctors on the cartesian closed E carries a strength derived from the two given strengths. We give it explicitly for the composit... |

1 |
Calculus of extensive quantities, arXiv [math.CT
- Kock
- 2011
(Show Context)
Citation Context ...der, we take the liberty to reason with “elements”. In particular, the “elements” of T (X) are simply called “distributions on X” The present text subsumes and simplifies the preliminary arXiv texts, =-=[12]-=-, [13], and has been presented in part in Krakow at the conference “Differential Geometry and Mathematical Physics, June-July 2011, in honour of Wlodzimierz Tulczyjew”, and at the Nancy Symposium “Set... |

1 |
Affine parts of monads
- Lindner
- 1979
(Show Context)
Citation Context ...d in [9] that this is equivalent to the assertion that for all X, Y , the map ψX,Y : T (X) × T (Y ) → T (X × Y ) is split monic with 〈T (pr1), T (pr2)〉 : T (X × Y ) → T (X) × T (Y ) as retraction. In =-=[15]-=-, it was proved that if E has finite limits, any commutative monad T has a maximal affine submonad T0, the “affine part of T ”. It is likewise a commutative monad. Speaking in elementwise terms, T0(X)... |

1 |
Kategorien und Funktoren, Teubner Stuttgart
- Pareigis
- 1969
(Show Context)
Citation Context ... even abelian groups). However, we shall need that such enrichment structure derives canonically from a certain property of the category. This is old wisdom, e.g. described in Pareigis’ 1969 textbook =-=[18]-=-, Section 4.1. We recall briefly the needed properties: Consider a category A with finite products and finite coproducts. If the unique map from the initial object to the terminal object is an isomorp... |

1 |
Algebraic Theories (revised version
- Wraith
(Show Context)
Citation Context ...ne submonad; probability distributions A strong monad T on a cartesian closed category E is called affine if T (1) = 1. For algebraic theories (monads on the category of sets), this was introduced in =-=[20]-=-. For strong monads, it was proved in [9] that this is equivalent to the assertion that for all X, Y , the map ψX,Y : T (X) × T (Y ) → T (X × Y ) is split monic with 〈T (pr1), T (pr2)〉 : T (X × Y ) → ... |