We would like to solve the following problem: find a mathematical model formulating I) quantum entanglement, II) particle-wave duality, III) universal objects (ur-sub-Planck objects): to be defined in terms of direct or inverse limits (defined by universal mapping properties) giving microcosm behaviors of space-time so as to give the smooth macrocosm space-time, and IV) the “curved” space-time associated with particles with mass in microcosm consistent with the notion of a light cone in macrocosm. Problems I) and II) are treated in Kato G., Europhys. Lett., 68 (2004) 467. In this paper, we will focus on III) and IV). As a candidate for such a model, we have introduced the category of presheaves over a site called a t-topos. During the last several years, the methods of category and sheaf theoretic approaches have been actively employed for the foundations of quantum physics and for quantum gravity. Particles, time, and space are presheafified in the following sense: a fundamental entity is a triple (m, κ, τ ) of presheaves so that for an object V in a t-site, a local datum (m(V ), κ(V ), τ(V )) may provide a local state of the particle m_ = m(V), i.e., the localization of presheaf m at V , in the neighborhood (κ(V ), τ(V )) of m_. By presheafifying matter, space, and time, t-topos can provide sheaf-theoretic descriptions of ur-entanglement and ur-particle and ur-wave states(1) formul ating the EPR-type non-locality and the duality in a double-slit experiment. Recall that presheaves m and m' are said to be ur-entangled when m and m' behave as one presheaf. Also recall: a presheaf m is said to be in particle ur-state (or wave ur-state) when the presheaf m is evaluated as m(V ) at a specified object V in the t-site (or when an object in the t-site is not specified). For more comments and the precise definitions of ur-entanglement and particle and wave ur-states, see the above-mentioned paper. The applications to a double-slit experiment and the EPR-type non-locality are described in detail in the forthcoming papers Kato G. and Tanaka T., Double slit experiment and t-topos, submitted to Found. Phys. and Kafatos M., Kato G., Roy S. and Tanaka T., The EPR-type non-locality and t-topos, to be submitted to Int. J. Pure Appl. Math., respectively. By the notion of decompositions of a presheaf and of an object of the t-site, ur-sub-Planck objects are defined as direct and inverse limits, respectively, in Definitions 2.1 and 2.4 in what will follow.



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