Related papers: Probability, valuations, hyperspace: Three monads …

Probability monads on categories of topological spaces are classical objects of study in the categorical approach to probability theory, with important applications in the semantics of probabilistic programming languages. We construct a…

We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps. We prove that every $\mathcal V_w$-algebra in our…

We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…

In recent times, there has been a growing interest in a structuralist understanding of probability, measure and integration theory. The present thesis contributes to this programme in three ways. First, we construct a commutative…

This paper is a follow-up to the author's work "Topology of probability measure space, I" devoted to investigation of the functors $\hat P$ and $P_\tau$ of spaces of probability $\tau$-smooth and Radon measures. In this part, we study the…

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In…

We show that there is weak distributive law of the Smyth hyperspace monad $\mathcal Q_{\mathsf V}$ (resp., the Hoare hyperspace monad $\mathcal H_{\mathsf V}$, resp. the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V}$ of quasi-lenses, resp.…

We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein…

We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes…

Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological…

The probabilistic powerdomain $\mathbf V X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X$, $\mathbf V X$ is again a quasi-continuous domain, and that the Scott and weak…

A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…

For a Tychonoff space $X$, the constructions $\hat P(X)$ and $P_\tau(X)$ of the spaces of probability Radon measures and probability $\tau$-smooth measures on $X$ are considered. It is proved that these constructions determine functors in…

Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_\mu(T)$, $\bar{h}_\mu(T)$ for any $\mu\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures…

We define a (preorder-enriched) category $\mathsf{Met}$ of quantale-valued metric spaces and uniformly continuous maps, with the essential requirement that the quantales are continuous. For each object $(X,d,Q)$ in this category, where $X$…

We examine conditions under which projective limits of topological spaces are preserved by the continuous valuation functor $\mathbf V$ and its subprobability and probability variants (used to represent probabilistic choice), by the Smyth…

This paper aims to use topological methods to compute $\mathrm{Ext}$ between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a…

It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with…

The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category…