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Jean Goubault-Larrecq

Publications and source records attributed to Jean Goubault-Larrecq.

At least 19 recordsLinked to original sources
arXiv2026

Semitopological Barycentric Algebras

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras. For example, the space of all continuous valuations (a very close cousin of measures) over a topological space is a topological cone, while probability valuations form a topological barycentric algebra, and subprobability valuations form a pointed topological barycentric algebra. Among other results, we show the existence of free semitopological cones over semitopological barycentric algebras and over pointed semitopological algebras, we investigate which semitopological barycentric algebras embed into semitopological cones and which pointed semitopological barycentric algebras embed strictly into semitopological cones. We study notions of local convexity, which split into weak local convexity, local convexity, local affineness and local linearity. We show that the weakly locally convex topological barycentric algebras are exactly the affine retracts of locally affine topological barycentric algebras. On locally convex barycentric algebras, we show sandwich theorems, extending theorems by Roth and Keimel on cones. A running theme of this paper is the notion of barycenters, which we progressively generalize until we reach a general notion of barycenters of continuous (resp., subprobability, probability) valuations, inspired by a definition of Choquet. We conclude with a general barycenter existence theorem, whose proof relies on the study of the Smyth poweralgebra, namely the topological barycentric algebra of all non-empty convex compact saturated subsets of a topological barycentric algebra.

math.FA↗
arXiv2026

Just Previsions

Previsions are positively homogeneous functionals, and are generalized forms of integration functionals. We investigate previsions -- just previsions, not sublinear or superlinear previsions as in previous work. We show that every prevision can be expressed as an infimum of sublinear previsions, and as a supremum of superlinear previsions under mild conditions. This extends to homeomorphisms between spaces of previsions and certain hyperspaces over spaces of sublinear or superlinear previsions, which can also be characterized in terms of orthogonality relations, making the construction a variant of a double powerspace construction.

cs.LO↗
arXiv2026

Stone Duality for Preordered Topological Spaces

A preordered topological space is a topological space with a preordering. We exhibit a Stone-like duality for preordered topological spaces, Inspired by a similar duality for bitopological spaces, due to Jung-Moshier and Jakl, and by a duality for preordered sets due to Bonsangue, Jacobs and Kok.

math.GN↗
arXiv2025

Distributing Retractions, Weak Distributive Laws and Applications to Monads of Hyperspaces, Continuous Valuations and Measures

Given two monads $S$, $T$ on a category where idempotents split, and a weak distributive law between them, one can build a combined monad $U$. Making explicit what this monad $U$ is requires some effort. When we already have an idea what $U$ should be, we show how to recognize that $U$ is indeed the combined monad obtained from $S$ and $T$: it suffices to exhibit what we call a distributing retraction of $ST$ onto $U$. We show that distributing retractions and weak distributive laws are in one-to-one correspondence, in a 2-categorical setting. We give three applications, where $S$ is the Smyth, Hoare or Plotkin hyperspace monad, $T$ is a monad of continuous valuations, and $U$ is a monad of previsions or of forks, depending on the case. As a byproduct, this allows us to describe the algebras of monads of superlinear, resp. sublinear previsions. In the category of compact Hausdorff spaces, the Plotkin hyperspace monad is sometimes known as the Vietoris monad, the monad of probability valuations coincides with the Radon monad, and we infer that the associated combined monad is the monad of normalized forks.

cs.LO↗
arXiv2025

Isomorphism Theorems between Models of Mixed Choice (Revised)

We relate the so-called powercone models of mixed non-deterministic and probabilistic choice proposed by Tix, Keimel, Plotkin, Mislove, Ouaknine, Worrell, Morgan, and McIver, to our own models of previsions. Under suitable topological assumptions, we show that they are isomorphic. We rely on Keimel's cone-theoretic variants of the classical Hahn-Banach separation theorems, using functional analytic methods, and on the Schröder-Simpson Theorem. Lemma 3.4 in the original 2017 version, published at MSCS, had a wrong proof, and we prove a repaired, albeit slightly less general version here.

cs.LO↗
arXiv2024

On the Preservation of Projective Limits by Functors of Non-Deterministic, Probabilistic, and Mixed Choice

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 hyperspace functor (demonic non-deterministic choice), by the Hoare hyperspace functor (angelic non-deterministic choice), by Heckmann's $\mathbf A$-valuation functor, by the quasi-lens functor, by the Plotkin hyperspace functor (erratic non-deterministic choice), and by prevision functors and powercone functors that implement mixtures of probabilistic and non-deterministic choice.

math.GN↗
arXiv2024

Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice

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. the monad $\mathcal P\ell_{\mathsf V}$ of lenses) over the continuous valuation monad $\mathbf V$, as well as over the subprobability valuation monad $\mathbf V_{\leq 1}$ and the probability valuation monad $\mathbf V_1$, on the whole category $\mathbf{Top}$ of topological spaces (resp., on certain full subcategories such as the category of locally compact spaces or of stably compact spaces). We show that the resulting weak composite monad is the author's monad of superlinear previsions (resp., sublinear previsions, resp. forks), possibly subnormalized or normalized depending on whether we consider $\mathbf V_{\leq 1}$ or $\mathbf V_1$ instead of $\mathbf V$. As a special case, we obtain a weak distributive law of the monad $\mathcal P\ell^{\mathrm q}_{\mathsf V} \cong \mathcal P\ell_{\mathsf V}$ over the monad of (sub)probability Radon measures $\mathbf R_\bullet$ on the category of stably compact spaces, which specializes further to a weak distributive laws of the Vietoris monad over $\mathbf R_\bullet$. The associated weak composite monad is the monad of (sub)normalized forks.

math.CT↗
arXiv2024

A cone-theoretic barycenter existence theorem

We show that every continuous valuation on a locally convex, locally convex-compact, sober topological cone $\mathfrak{C}$ has a barycenter. This barycenter is unique, and the barycenter map $β$ is continuous, hence is the structure map of a $\mathbf V_{\mathrm w}$-algebra, i.e., an Eilenberg-Moore algebra of the extended valuation monad on the category of $T_0$ topological spaces; it is, in fact, the unique $\mathbf V_{\mathrm w}$-algebra that induces the cone structure on $\mathfrak{C}$.

math.GN↗
arXiv2024

A Few Projective Classes of (Non-Hausdorff) Topological Spaces

A class of topological spaces is projective (resp., $ω$-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, ($ω$-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even $ω$-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.

math.GN↗
arXiv2023

A few characterizations of topological spaces with no infinite discrete subspace

We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erdös and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.

math.GN↗
arXiv2023

A Radon-Nikodým Theorem for Valuations

We enquire under which conditions, given two $σ$-finite, $ω$-continuous valuations $ν$ and $μ$, $ν$ has density with respect to $μ$. The answer is that $ν$ has to be absolutely continuous with respect to $μ$, plus a certain Hahn decomposition property, which happens to be always true for measures.

math.FA↗
arXiv2023

Continuous R-valuations

We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Théron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $τ$-smooth measure $μ$ (defined on the Borel $σ$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $μ$, when the support of $μ$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

math.GN↗
arXiv2023

Statures and Sobrification Ranks of Noetherian Spaces

There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces. The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature $||X||$ of a Noetherian space $X$ as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space $X$, of the space of finite multisets on $X$, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces. The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020). We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on $X$ with the so-called prefix topology. Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their sobrification ranks.

math.GN↗
arXiv2022

Complete Quasi-Metrics for Hyperspaces, Continuous Valuations, and Previsions

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi-metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda-complete, and that the associated topology is the weak topology. We do this not only for probability distributions, represented as normalized continuous valuations, but also for subprobability distributions, for various hyperspaces, and in general for different brands of functionals. Those functionals model probabilistic choice, angelic and demonic non-deterministic choice, and their combinations. The mathematics needed for those results are more demanding than in the simpler case of metric spaces. To obtain our results, we prove a few other results that have independent interest, notably: continuous Yoneda-complete spaces are consonant; on a continuous Yoneda-complete space, the Scott topology on the space of $\overline{\mathbb{R}}_+$-valued lower semicontinuous maps coincides with the compact-open and Isbell topologies, and the subspace topology on spaces of $α$-Lipschitz continuous maps also coincides with the topology of pointwise convergence, and is stably compact; we introduce and study the so-called Lipschitz-regular quasi-metric spaces, and we show that the formal ball functor induces a Kock-Zöberlein monad, of which all algebras are Lipschitz-regular; and we prove a minimax theorem where one of the spaces is not compact Hausdorff, but merely compact.

math.GN↗
arXiv2022

On Weakly Hausdorff Spaces and Locally Strongly Sober Spaces

We show that the locally strongly sober spaces are exactly the coherent sober spaces that are weakly Hausdorff in the sense of Keimel and Lawson. This allows us to describe their Stone duals explicitly. As another application, we show that weak Hausdorffness is a sufficient condition for lenses and of quasi-lenses to form homeomorphic spaces, generalizing previously known results.

math.GN↗
arXiv2021

A Domain-Theoretic Approach to Statistical Programming Languages

We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin' s monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations. (Implementations by themselves are not addressed here.) Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: all measures on the real line can be modeled by minimal valuations on the domain $\mathbf{I}\mathbb{R}_\bot$ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples in order to demonstrate that our semantics computes exactly as one would expect, and in order to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance. Our examples also include some useful, non-trivial cases of distributions on higher-order objects.

cs.LO↗
arXiv2021

Separating minimal valuations, point-continuous valuations and continuous valuations

We give two concrete examples of continuous valuations on dcpo's to separate minimal valuations, point-continuous valuations and continuous valuations: (1) Let $\mathcal J$ be the Johnstone's non-sober dcpo, and $μ$ be the continuous valuation on $\mathcal J$ with $μ(U) =1$ for nonempty Scott opens $U$ and $μ(U) = 0$ for $U=\emptyset$. Then $μ$ is a point-continuous valuation on $\mathcal J$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb R_{l}$. Its restriction to the open subsets of $\mathbb R_{l}$ is a continuous valuation $λ$. Then its image valuation $\overlineλ$ through the embedding of $\mathbb R_{l}$ into its Smyth powerdomain $\mathcal Q\mathbb R_{l}$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $\overlineλ$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo's.

cs.LO↗