2,465 research outputs found

    Combinatorial structure of type dependency

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    We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf category. The objects of the presheaf category encode the basic judgements of a dependent sequent calculus, while the action of the monad encodes the deduction rules; so by giving an explicit description of the monad, we obtain an explicit account of the combinatorics of type dependency. We find that this combinatorics is controlled by a particular kind of decorated ordered tree, familiar from computer science and from innocent game semantics. Furthermore, we find that the monad at issue is of a particularly well-behaved kind: it is local right adjoint in the sense of Street--Weber. In future work, we will use this fact to describe nerves for dependent type theories, and to study the coherence problem for dependent type theory using the tools of two-dimensional monad theory.Comment: 35 page

    Syntactic presentations for glued toposes and for crystalline toposes

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    We regard a geometric theory classified by a topos as a syntactic presentation for the topos and develop tools for finding such presentations. Extensions (or expansions) of geometric theories, which can not only add axioms but also symbols and sorts, are treated as objects in their own right, to be able to build up complex theories from parts. The role of equivalence extensions, which leave the theory the same up to Morita equivalence, is investigated. Motivated by the question what the big Zariski topos of a non-affine scheme classifies, we show how to construct a syntactic presentation for a topos if syntactic presentations for a covering family of open subtoposes are given. For this, we introduce a transformation of theory extensions such that when the result, dubbed a conditional extension, is added to a theory, it requires part of the data a model is made of only under some condition given in the form of a closed geometric formula. We also give a general definition for systems of interdependent theory extensions, to be able to talk about compatible syntactic presentations not only for the open subtoposes in a given cover but also for their finite intersections. An important concept for finding classified theories of toposes in concrete situations is that of theories of presheaf type. We develop several techniques for extending a theory while preserving the presheaf type property, and give a list of examples of simple extensions which can destroy it. Finally, we determine a syntactic presentation of the big crystalline topos of a scheme. In the case of an affine scheme, this is accomplished by showing that the biggest part of the classified theory is of presheaf type and transforming the site defining the crystalline topos into the canonical presheaf site for this theory, while the remaining axioms induce the Zariski topology. Then we can apply our results on gluing classifying toposes to obtain a classified theory even in the non-affine case

    A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory

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    This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf--the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study `truth objects' in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.Comment: 34 pages, no figure

    On the geometric theory of local MV-algebras

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    We investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MV-algebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.Comment: 52 page

    A Topos Foundation for Theories of Physics: I. Formal Languages for Physics

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    This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.Comment: 36 pages, no figure
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