Sheaves in Geometry and Logic

Lane, Saunders Mac; Moerdijk, Ieke

doi:10.1007/978-1-4612-0927-0

Cited by 597 publications

(174 citation statements)

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“…Now, the category 2 1 op Set of contravariant functors from 2 1 to Set (morphisms are natural transformations) is isomorphic to the category of precubical sets. This implies, by general theorems [44], that Υ S is an elementary topos. Moreover it is complete and co-complete because Set is complete and co-complete.…”

Section: Interpretation In Terms Of Concurrency Theorymentioning

confidence: 69%

Some geometric perspectives in concurrency theory

Goubault

2003

Homology, Homotopy and Applications

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Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of "geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the "direction" of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: "directed" or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ω-categorical and topological techniques.

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Section: Interpretation In Terms Of Concurrency Theorymentioning

confidence: 69%

Some geometric perspectives in concurrency theory

Goubault

2003

Homology, Homotopy and Applications

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“…The qr-numbers contains the standard reals, have orders < and ≤ compatible with those on the standard rationals but which are not total, form a field under + and × in the restricted sense that if a number does not have an inverse then it is zero, and has a distance function that defines a metric topology on the qr-numbers with respect to which the rationals are dense [21,23,25].…”

Section: Qr-numbers For a Massive Galilean Particlementioning

confidence: 99%

“…The mathematical construction underpinning the construction of qr-numbers is that of a topos [8,21]. We use a spatial topos, Shv(X), the category of sheaves on a topological space X.…”

Section: Qr-numbers For a Massive Galilean Particlementioning

confidence: 99%

“…Therefore, for any open subset U of E S , the function a Q restricted to U plays the role of a qr-number defined to extent U . The standard real numbers on U are represented by locally constant functions [21] that include the constant function cI Q (U ) defined for c ∈ R andÎ the identity operator. The qr-numbers contain all the continuous functions, such as polynomials, of the functions a Q .…”

Section: Qr-numbers For a Massive Galilean Particlementioning

confidence: 99%

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Spatial Localization in Quantum Theory Based on qr-numbers

Corbett

Durt²

2010

Found Phys

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We show how trajectories can be reintroduced in quantum mechanics provided that its spatial continuum is modelled by a variable real number (qr-number) continuum. Such a continuum can be constructed using only standard Hilbert space entities. In this approach, the geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare trajectories in this new description of space-time with the corresponding Bohmian picture.

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“…Recall [SGA4,MM,... ] that a topos is (a category equivalent to) the category of sheaves on a small site. More explicitly, a site is a pair (C, J), where C is a small category with pullbacks, while J assigns to each object C in C a collection J(C) of "covering families" satisfying certain axioms:…”

Section: Preliminary Definitionsmentioning

confidence: 99%