2,465 research outputs found
Combinatorial structure of type dependency
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
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
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
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
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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