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Geometric and higher order logic in terms of abstract Stone duality
 THEORY AND APPLICATIONS OF CATEGORIES
, 2000
"... The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this ..."
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The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
Subspaces in abstract Stone duality
 Theory and Applications of Categories
, 2002
"... ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idemp ..."
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ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The paper is largely concerned with the construction of such a category out of one that merely has powers of some fixed object Σ. It builds on Sober Spaces and Continuations, where the related but weaker notion of abstract sobriety was considered. The construction is done first by formally adjoining certain equalisers that Σ (−) takes to coequalisers, then using Eilenberg–Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory. The comprehension calculus has a normalisation theorem, by which every type can
The Demonic Product of Probabilistic Relations
, 2001
"... The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the nondeterministic fringe of the probabilistic relations behaves properly: the ..."
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The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the nondeterministic fringe of the probabilistic relations behaves properly: the fringe of the product equals the demonic product of the fringes.
ON THE AXIOMATISATION OF BOOLEAN CATEGORIES WITH AND WITHOUT MEDIAL
"... should be used for describing an object that ..."
FirstOrder Logical Duality
, 2008
"... Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is re ..."
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Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is replaced by an embedding of a Boolean coherent category, B, into a topos of equivariant sheaves on a topological groupoid of setvalued models and isomorphisms between them. The latter is a groupoid representation of the topos of coherent sheaves on B, analogously to how the Stone space of a Boolean algebra is a spatial representation of the ideal completion of the algebra, and the category B can then be recovered from its semantical groupoid, up to pretopos completion. By equipping the groupoid of sets and bijections with a particular topology, one obtains a particular topological groupoid which plays a role analogous to that of the discrete space 2, in being the dual of the object classifier and the object one ‘homs into ’ to recover a Boolean coherent category from its semantical groupoid. Both parts of the adjunction, then, consist of ‘homming into sets’, similarly to how both parts of the equivalence between Boolean algebras and Stone spaces consist of ‘homming into 2’. By slicing over this groupoid (modified to display an alternative setup), Chapter 3 shows how the adjunction specializes to the case of firstorder single sorted logic to yield an adjunction between such theories and an independently characterized slice category of topological groupoids such that the counit component at a theory is an isomorphism. Acknowledgements I would like, first and foremost, to thank my supervisor Steve Awodey. I would like to thank the members of the committee: Jeremy Avigad, Lars
Proof of the Decidability of the Uniform Word Problem for Monads Assisted by Elf
, 1994
"... . We present a full proof of a canonical system for adjunctions as already suggested in [Cur93]. Termination can be shown in a similar style to [HL86]. Confluence is shown by checking all critical pairs. This is done firstly in the Larch Prover [GG91] and secondly more correctly in the programming l ..."
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. We present a full proof of a canonical system for adjunctions as already suggested in [Cur93]. Termination can be shown in a similar style to [HL86]. Confluence is shown by checking all critical pairs. This is done firstly in the Larch Prover [GG91] and secondly more correctly in the programming language Elf [Pfe89]. Exploiting theorems from category theory [BW85] this system can be used to solve the uniform word problem for monads. The resulting decision procedure is finally implemented in Elf. 1 Introduction A general problem in category theory is to check the commutativity of certain diagrams where diagrams are nothing but a compact encoding and visualization of equations involving morphisms. In [FS90] one can even find a suitable graphical language. Checking the commutativity means to check the equality of morphisms in a given category. To support this process one can solve the uniform word problem for this category which is of course not always possible having the halting probl...
Invariants of boundary link cobordism II: The Blanchfield–Duval form
 LMS LECTURE NOTE SER. 330
, 2006
"... We use the BlanchfieldDuval form to define complete invariants for the cobordism group C2q−1(Fµ) of (2q − 1)dimensional µcomponent boundary links (for q ≥ 2). The author solved the same problem in earlier work via Seifert forms. Although Seifert forms are convenient in explicit computations, the ..."
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We use the BlanchfieldDuval form to define complete invariants for the cobordism group C2q−1(Fµ) of (2q − 1)dimensional µcomponent boundary links (for q ≥ 2). The author solved the same problem in earlier work via Seifert forms. Although Seifert forms are convenient in explicit computations, the BlanchfieldDuval form is more intrinsic and appears naturally in homology surgery theory. The free cover of the complement of a link is constructed by pasting together infinitely many copies of the complement of a µcomponent Seifert surface. We prove that the algebraic analogue of this construction, a functor denoted B, identifies the author’s earlier invariants with those defined here. We show that B is equivalent to a universal localization of categories and describe the structure of the modules sent to zero. Taking coefficients in a semisimple Artinian ring, we deduce that the Witt group of Seifert forms is isomorphic to the Witt group of BlanchfieldDuval forms.
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
Network as a computer: ranking paths to find flows
, 802
"... Abstract. We explore a simple mathematical model of network computation, based on Markov chains. Similar models apply to a broad range of computational phenomena, arising in networks of computers, as well as in genetic, and neural nets, in social networks, and so on. The main problem of interaction ..."
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Abstract. We explore a simple mathematical model of network computation, based on Markov chains. Similar models apply to a broad range of computational phenomena, arising in networks of computers, as well as in genetic, and neural nets, in social networks, and so on. The main problem of interaction with such spontaneously evolving computational systems is that the data are not uniformly structured. An interesting approach is to try to extract the semantical content of the data from their distribution among the nodes. A concept is then identified by finding the community of nodes that share it. The task of data structuring is thus reduced to the task of finding the network communities, as groups of nodes that together perform some nonlocal data processing. Towards this goal, we extend the ranking methods from nodes to paths, which allows us to extract information about the likely flow biases from the available static information about the network. 1
Coherence for categorified operadic theories
"... It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) ..."
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It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) theory P, and show that every weak Pcategory is equivalent via Pfunctors and Ptransformations to a strict Pcategory. This strictification functor is then shown to have an interesting universal property. 1