104 research outputs found
Frames and Topological Algebras for a Double-Power Monad
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
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
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
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
The Expectation Monad in Quantum Foundations
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
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
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
For a (possibly large) realized limit sketch such that every
-model is small in a suitable sense we show that the category of
cocontinuous functors into a
cocomplete category is equivalent to the category
of -valued
-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 , study the corresponding notion of an
internal net-based topological space object, and deduce from our main result
that cocontinuous functors into a cocomplete
category correspond to net-based cotopological space objects
internal to . Finally we describe a limit sketch which models
and deduce from our main result that continuous
functors into a complete category
correspond to frame-based topological space objects internal to .
Thus, we characterize both as a cocomplete and as a complete
category. Thereby we get two new conceptual proofs of Isbell's classification
of cocontinuous functors 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
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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