Goro Katohttps://works.bepress.com/gkato/Recent works by Goro Katoen-usCopyright (c) 2019 All rights reserved.Thu, 01 Mar 2012 08:00:00 +00003600Temporal Topos and U-Singularitieshttps://works.bepress.com/gkato/17/Several papers and books by C. Isham, C.Isham-A. Doering, F. Van Oystaeyen, A.Mallios-I. Raptis, C. Mulvey, and Guts and Grinkevich, have been published on the methods of categories and sheaves to study quantum gravity. Needless to say, there are well-written treatises on quantum gravity whose methods are non-categorical and non-sheaf theoretic. This paper may be one of the first papers explaining the methods of sheaves with minimally required background that retains experimental applications. Temporal topos (t-topos) is related to the topos approach to quantum gravity being developed by Prof. Chris Isham of the Oxford-Imperial research group (with its foundations inthe work of F. W. Lawvere). However, in spite of strong influence from papers by Isham, our method of t-topos is much more direct in the following sense. Our approach is much closer to the familiar applications of the original algebraic geometric topos where little logic is involved.Thu, 01 Mar 2012 08:00:00 +0000https://works.bepress.com/gkato/17/ArticlesUrcohomologies and Cohomologies of N -Complexeshttps://works.bepress.com/gkato/11/For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.Fri, 01 Jan 2010 08:00:00 +0000https://works.bepress.com/gkato/11/Articles<em>u</em>-Singularity and <em>t</em>-Topos Theoretic Entropyhttps://works.bepress.com/gkato/6/We will give descriptions of u-singularities as we introduce the notion of t-topos theoretic entropies. The unifying methodology for a u-singularity is the universal mapping property of an inverse or direct limit. The qualitative, conceptual, and structural analyses of u-singularities are given in terms of inverse and direct limits of micro decompositions of a presheaf and coverings of an object in t-site in the theory of temporal topos.Fri, 01 Jan 2010 08:00:00 +0000https://works.bepress.com/gkato/6/ArticlesMicrocosm to Macrocosm Via the Notion of a Sheaf (Observers in Terms of t-Topos)https://works.bepress.com/gkato/9/The fundamental approach toward matter, space and time is that particles (either objects of macrocosm or microcosm), space and time are all presheafified. Namely, the concept of a presheaf is most fundamental for matter, space and time. An observation of a particle is represented by a morphism from the observed particle (its associated presheaf) to the observer (its associated presheaf) over a specified object (called a generalized time period) of a t-site (i.e. a category with a Grothendieck topology). This formulation provides a scale independent and background space-time free theory (since, for the t-topos theoretic formulation, space and time are discretely defined by the associated particle, whose particle-dependency is a consequence of quantum entanglement.). It is our basic scheme that the method of t-topos may provide a device for understanding and concrete formulation of macro-object and micro-object interconnection as morphisms in the sense of t-topos.Goro KatoTue, 01 Jan 2008 08:00:00 +0000https://works.bepress.com/gkato/9/ArticlesSheaf Theoretic Formulation of Entanglementhttps://works.bepress.com/gkato/3/A formulation in terms of sheaf theoretic (or categorical) notions for quantum entanglement is given with direct experimental consequences. The notions from sheaf theory and category theory give structural theory, i.e., qualitative theory, as a candidate for quantum gravity. Its advantage is the following: it provides not only space-time background independent, but also scale independent.This theory is called the theory of temporal topos (or simply t-topos theory).Goro C. KatoTue, 01 Jan 2008 08:00:00 +0000https://works.bepress.com/gkato/3/ArticlesDouble-Slit Interference and Temporal Toposhttps://works.bepress.com/gkato/7/The electron double-slit interference is re-examined from the point of view of temporal topos. Temporal topos (or t-topos) is an abstract algebraic (categorical) method using the theory of sheaves. A brief introduction to t-topos is given. When the structural foundation for describing particles is based on t-topos, the particle-wave duality of electron is a natural consequence. A presheaf associated with the electron represents both particle-like and wave-like properties depending upon whether an object in the site (t-site) is specified (particle-like) or not (wave-like). It is shown that the localization of the electron at one of the slits is equivalent to choosing a particular object in the t-site and that the electron behaves as a wave when it passes through a double-slit because there are more than one object in the t-site. Also, the single-slit diffraction is interpreted as a result of the possibility of many different ways of factoring a morphism between two objects.Goro Kato et al.Wed, 01 Nov 2006 08:00:00 +0000https://works.bepress.com/gkato/7/ArticlesElemental t.g. Principles of Relativistic t-Topos (Presheafification of Matter, Space, and Time)https://works.bepress.com/gkato/1/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.Fri, 01 Jul 2005 07:00:00 +0000https://works.bepress.com/gkato/1/ArticlesElemental principles of <em>t</em>-toposhttps://works.bepress.com/gkato/8/In this paper, a sheaf-theoretic approach toward fundamental problems in quantum physics is made. For example, the particle-wave duality depends upon whether or not a presheaf is evaluated at a specified object. The t-topos theoretic interpretations of double-slit interference, uncertainty principle(s), and the EPR-type non-locality are given. As will be explained, there are more than one type of uncertainty principle: the absolute uncertainty principle coming from the direct limit object corresponding to the refinements of coverings, the uncertainty coming from a micromorphism of shortest observable states, and the uncertainty of the observation image. A sheaf theoretic approach for quantum gravity has been made by Isham-Butterfield in (Found. Phys. 30 (2000) 1707), and by Raptis based on abstract differential geometry in Mallios A. and Raptis I. Int. J. Theor. Phys. 41 (2002), qr-qc/0110033; Mallios A. Remarks on "singularities" (2002) qr-qc/0202028; Mallios A. and Raptis I. Int. J. Theor. Phys. 42 (2003) 1479, qr-qc/0209048. See also the preprint The translocal depth-structure of space-time, Connes' "Points, Speaking to Each Other", and the (complex) structure of quantum theory, for another approach relevant to ours. Special axioms of t-topos formulation are: i) the usual linear-time concept is interpreted as the image of the presheaf (associated with time) evaluated at an object of a t-site (i.e., a category with a Grothendieck topology). And an object of this t-site, which is said to be a generalized time period, may be regarded as a hidden variable and ii) every object (in a particle ur-state) of microcosm (or of macrocosm) is regarded as the microcosm (or macrocosm) component of a product category for a presheaf evaluated at an object in the t-site. The fundamental category Ŝ is defined as the category of πα ∈ Δ Cα-valued presheaves on the t-site S, where Δ is an index set. The study of topological properties of S with respect to the nature of multi-valued presheaves is left for future study on the t-topos version of relativity (see , On t.g. Principles of relativistic t-topos, in preparation; Guts A. K. and Grinkevich E. B. Toposes in General Theory of Relativity (1996), arXiv:gr-qc/9610073, 31). We let C1 and C2 be microcosm and macrocosm discrete categories, respectively, in what will follow. For further development see also Kato G. Presheafification of Matter, Space and Time, International Workshop on Topos and Theoretical Physics, July 2003, Imperial College (invited talk, 2003).Goro KatoMon, 01 Nov 2004 08:00:00 +0000https://works.bepress.com/gkato/8/ArticlesCategory Theory and Consciousnesshttps://works.bepress.com/gkato/14/Tue, 01 Jan 2002 08:00:00 +0000https://works.bepress.com/gkato/14/Contributions to BooksSheaf Cohomology of Conscious Entityhttps://works.bepress.com/gkato/15/Awareness of a conscious entity can exist without elements; therefore, the general notion of an object of a category is employed. One of the characterization of understanding is: for a given local infonnation (awareness) there exists a global information whose restriction is the given information. For such mental activities, category and sheaf theories are employed to formulate consciousness. We will show that the cohomology (more general precohomology) object, a subquotient object, better represents the essence of a conscious entity than an object itself. We will also give a definition of an observation to fonnulate the collapse of the wave and the wave property.Goro KatoTue, 01 Jan 2002 08:00:00 +0000https://works.bepress.com/gkato/15/Conference Proceedings