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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.12333">arXiv:2411.12333</a> <span> [<a href="https://arxiv.org/pdf/2411.12333">pdf</a>, <a href="https://arxiv.org/format/2411.12333">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Correspondences between codensity and coupling-based liftings, a practical approach </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Humeau%2C+S">Samuel Humeau</a>, <a href="/search/math?searchtype=author&query=Petrisan%2C+D">Daniela Petrisan</a>, <a href="/search/math?searchtype=author&query=Rot%2C+J">Jurriaan Rot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.12333v3-abstract-short" style="display: inline;"> The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0, 1], and as an infimum, based on probabilistic couplings.Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12333v3-abstract-full').style.display = 'inline'; document.getElementById('2411.12333v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.12333v3-abstract-full" style="display: none;"> The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0, 1], and as an infimum, based on probabilistic couplings.Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form of codensity liftings and what we refer to as coupling-based liftings. Both lift endofunctors on the category Set of sets and functions to that of pseudometric spaces, and both are parameterised by modalities from coalgebraic modal logic.A generalisation of the Kantorovich-Rubinstein duality has been more nebulous-it is known not to work in some cases. In this paper we propose a compositional approach for obtaining such generalised dualities for a class of functors, which is closed under coproducts and products. Our approach is based on an explicit construction of modalities and also applies to and extends known cases such as that of the powerset functor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12333v3-abstract-full').style.display = 'none'; document.getElementById('2411.12333v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.13489">arXiv:2106.13489</a> <span> [<a href="https://arxiv.org/pdf/2106.13489">pdf</a>, <a href="https://arxiv.org/ps/2106.13489">ps</a>, <a href="https://arxiv.org/format/2106.13489">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4204/EPTCS.351.14">10.4204/EPTCS.351.14 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Semialgebras and Weak Distributive Laws </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Petri%C5%9Fan%2C+D">Daniela Petri艧an</a>, <a href="/search/math?searchtype=author&query=Sarkis%2C+R">Ralph Sarkis</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.13489v3-abstract-short" style="display: inline;"> Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.13489v3-abstract-full').style.display = 'inline'; document.getElementById('2106.13489v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.13489v3-abstract-full" style="display: none;"> Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.13489v3-abstract-full').style.display = 'none'; document.getElementById('2106.13489v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">In Proceedings MFPS 2021, arXiv:2112.13746</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> EPTCS 351, 2021, pp. 218-241 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.08841">arXiv:1702.08841</a> <span> [<a href="https://arxiv.org/pdf/1702.08841">pdf</a>, <a href="https://arxiv.org/ps/1702.08841">ps</a>, <a href="https://arxiv.org/format/1702.08841">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Formal Languages and Automata Theory">cs.FL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/S0960129521000074">10.1017/S0960129521000074 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantifiers on languages and codensity monads </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gehrke%2C+M">Mai Gehrke</a>, <a href="/search/math?searchtype=author&query=Petrisan%2C+D">Daniela Petrisan</a>, <a href="/search/math?searchtype=author&query=Reggio%2C+L">Luca Reggio</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.08841v3-abstract-short" style="display: inline;"> This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.08841v3-abstract-full').style.display = 'inline'; document.getElementById('1702.08841v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.08841v3-abstract-full" style="display: none;"> This paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.08841v3-abstract-full').style.display = 'none'; document.getElementById('1702.08841v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages. Presentation improved and details of several proofs added. The main results are unchanged</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.1.1; F.4.1; F.4.3 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Math. Struct. Comp. Sci. 30 (2020) 1054-1088 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.08264">arXiv:1603.08264</a> <span> [<a href="https://arxiv.org/pdf/1603.08264">pdf</a>, <a href="https://arxiv.org/format/1603.08264">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Formal Languages and Automata Theory">cs.FL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> The Sch眉tzenberger product for syntactic spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gehrke%2C+M">Mai Gehrke</a>, <a href="/search/math?searchtype=author&query=Petrisan%2C+D">Daniela Petrisan</a>, <a href="/search/math?searchtype=author&query=Reggio%2C+L">Luca Reggio</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1603.08264v1-abstract-short" style="display: inline;"> Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to a setting that is well-suited for applying existing tools from Stone duality as applied in semantics. The main focus of the paper is th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.08264v1-abstract-full').style.display = 'inline'; document.getElementById('1603.08264v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.08264v1-abstract-full" style="display: none;"> Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to a setting that is well-suited for applying existing tools from Stone duality as applied in semantics. The main focus of the paper is the development of topo-algebraic constructions pertinent to the treatment of languages given by logic formulas. In particular, using the standard semantic view of quantification as projection, we derive a notion of Sch眉tzenberger product for Boolean spaces with internal monoids. This makes heavy use of the Vietoris construction, and its dual functor, which is central to the coalgebraic treatment of classical modal logic. We show that the unary Sch眉tzenberger product for spaces, when applied to a recogniser for the language associated to a formula with a free first-order variable, yields a recogniser for the language of all models of the corresponding existentially quantified formula. Further, we generalise global and local versions of the theorems of Sch眉tzenberger and Reutenauer characterising the languages recognised by the binary Sch眉tzenberger product. Finally, we provide an equational characterisation of Boolean algebras obtained by local Sch眉tzenberger product with the one element space based on an Egli-Milner type condition on generalised factorisations of ultrafilters on words. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.08264v1-abstract-full').style.display = 'none'; document.getElementById('1603.08264v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.1.1; F.4.1; F.4.3 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1210.1433">arXiv:1210.1433</a> <span> [<a href="https://arxiv.org/pdf/1210.1433">pdf</a>, <a href="https://arxiv.org/ps/1210.1433">ps</a>, <a href="https://arxiv.org/format/1210.1433">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Relation Liftings on Preorders and Posets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bilkova%2C+M">Marta Bilkova</a>, <a href="/search/math?searchtype=author&query=Kurz%2C+A">Alexander Kurz</a>, <a href="/search/math?searchtype=author&query=Petrisan%2C+D">Daniela Petrisan</a>, <a href="/search/math?searchtype=author&query=Velebil%2C+J">Jiri Velebil</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1210.1433v1-abstract-short" style="display: inline;"> The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.1433v1-abstract-full').style.display = 'inline'; document.getElementById('1210.1433v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1210.1433v1-abstract-full" style="display: none;"> The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss's coalgebraic over posets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.1433v1-abstract-full').style.display = 'none'; document.getElementById('1210.1433v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1006.3027">arXiv:1006.3027</a> <span> [<a href="https://arxiv.org/pdf/1006.3027">pdf</a>, <a href="https://arxiv.org/ps/1006.3027">ps</a>, <a href="https://arxiv.org/format/1006.3027">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> Algebraic Theories over Nominal Sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kurz%2C+A">Alexander Kurz</a>, <a href="/search/math?searchtype=author&query=Petri%C5%9Fan%2C+D">Daniela Petri艧an</a>, <a href="/search/math?searchtype=author&query=Velebil%2C+J">Ji艡铆 Velebil</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1006.3027v1-abstract-short" style="display: inline;"> We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1006.3027v1-abstract-full').style.display = 'inline'; document.getElementById('1006.3027v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1006.3027v1-abstract-full" style="display: none;"> We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to recover the logics of Gabbay-Mathijssen and Clouston-Pitts, and apply classical results from universal algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1006.3027v1-abstract-full').style.display = 'none'; document.getElementById('1006.3027v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 June, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 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