Elemental t . g . principles of relativistic t - topos ( ∗ ) ( Presheafification of matter

– We would like to solve the following problem: find a mathematical model formu­ lating 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 proper­ ties) 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. Par­ ticles, 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() formulating the EPR-type non-locality and the duality in a double-slit experiment. Recall that 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., 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 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.