Related papers: Categorical algebra of conditional probability

We present a novel, yet rather simple construction within the traditional framework of Scott domains to provide semantics to probabilistic programming, thus obtaining a solution to a long-standing open problem in this area. Unlike current…

Batanin and Leinster's work on globular operads has provided one of many potential defnitions of a weak $\omega$-category. Through the language of globular operads they construct a monad whose algebras encode weak $\omega$-categories. The…

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^{\alpha} & \Mor_\A…

Strong bisimulation for labelled transition systems is one of the most fundamental equivalences in process algebra, and has been generalised to numerous classes of systems that exhibit richer transition behaviour. Nearly all of the ensuing…

We provide the expected constructions of weakly $\omega$-categorified models (in the sense of Bressie) of the theory of groups and quandles which arise by replacing the homotopies used to give equivalence relations in the theory of…

This paper presents categorical structures on classical measure spaces and quantum measure spaces in order to deal with canonical maps associated with conditional measures as morphisms. We extend the Riesz-Markov-Kakutani representation…

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties…

It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for…

Two high-level "pictures" of probability theory have emerged: one that takes as central the notion of random variable, and one that focuses on distributions and probability channels (Markov kernels). While the channel-based picture has been…

We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…

We recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with…

We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are…

Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful…

Quantitative algebras are $\Sigma$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations.…

Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a…

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps…

The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a…

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are…