104 research outputs found

    Frames and Topological Algebras for a Double-Power Monad

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    We study the algebras for the double power monad on the Sierpinski space in the Cartesian closed category of equilogical spaces and produce a connection of the algebras with frames. The results hint at a possible synthetic, constructive approach to frames via algebras, in line with that considered in Abstract Stone Duality by Paul Taylor and others

    A Recipe for State-and-Effect Triangles

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    In the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains

    Enriched Stone-type dualities

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    A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [0,1][0,1] and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in [0,1][0,1]

    The Expectation Monad in Quantum Foundations

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    The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.Comment: In Proceedings QPL 2011, arXiv:1210.029

    Moduli stacks of algebraic structures and deformation theory

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    We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.Comment: several corrections, especially in sections 6 and 7. Final version, to appear in the J. Noncommut. Geo

    Positivity relations on a locale

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    This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)

    Large limit sketches and topological space objects

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    For a (possibly large) realized limit sketch S\mathcal{S} such that every S\mathcal{S}-model is small in a suitable sense we show that the category of cocontinuous functors Mod(S)→C\mathrm{Mod}(\mathcal{S}) \to \mathcal{C} into a cocomplete category C\mathcal{C} is equivalent to the category ModC(Sop)\mathrm{Mod}_{\mathcal{C}}(\mathcal{S}^{\mathrm{op}}) of C\mathcal{C}-valued Sop\mathcal{S}^{\mathrm{op}}-models. From this result we deduce universal properties of several examples of cocomplete categories appearing in practice. It can be applied in particular to infinitary Lawvere theories, generalizing the well-known case of finitary Lawvere theories. We also look at a large limit sketch which models Top\mathsf{Top}, study the corresponding notion of an internal net-based topological space object, and deduce from our main result that cocontinuous functors Top→C\mathsf{Top} \to \mathcal{C} into a cocomplete category C\mathcal{C} correspond to net-based cotopological space objects internal to C\mathcal{C}. Finally we describe a limit sketch which models Topop\mathsf{Top}^{\mathrm{op}} and deduce from our main result that continuous functors Top→C\mathsf{Top} \to \mathcal{C} into a complete category C\mathcal{C} correspond to frame-based topological space objects internal to C\mathcal{C}. Thus, we characterize Top\mathsf{Top} both as a cocomplete and as a complete category. Thereby we get two new conceptual proofs of Isbell's classification of cocontinuous functors Top→Top\mathsf{Top} \to \mathsf{Top} in terms of topological topologies.Comment: 42 pages; comments are appreciated, in particular if detailed proofs of the 'folklore results' already appear elsewher

    A Universal Characterization of the Double Powerlocale

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    This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321. The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc
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