Temporal Topos and U-Singularities

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 in the 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. The distinguishable aspects of this paper “Temporal Topos and U-Singularity” from other topos theorists’ appoaches are the following. For a particle, we consider a presheaf associated with the particle. By definition, a presheaf is a contravarinat functor; however, in the t-topos theory, such a presheaf need not be defined for every object of a t-site over which the topos of presheaves are defined. When such an associated presheaf is not defined (or non-reified), we say that the presheaf (the particle) is in ur-wave state. Therefore, the duality is already embedded in our t-topos theory. We also have the notion of a (micro) decomposition of a presheaf (a particle) to obtain microcosm objects. Another important aspect of our approach is the associated space and time sheaves for a given particle-presheaf. The sheaves associated with space, time, and space-time are treated differently from a particle associated presheaf. Namely, Yoneda Lemma and its embedding are crucial for formulating and capturing the nature of space-time. In this formulation, the space and time sheaves would not exist unless a particle (presheaf) exists. Such a non-locality nature as the EPR type non-locality is also embedded in t-topos. Applications to singularities (a big bang, black holes, and subplanck objects) are formulated in terms of universal mapping properties of direct limit and inverse limit in category theory. Furthermore, the uncertainty principle is formulated through the concept of a micro-morphism in t-site. Our t-topos theoretic approach enables us to formulate a light cone in macrocosm and also in microcosm. However, such a light cone in microcosm has non-reified space-time regions because of the uncertainty principle (a miro­ morophsim).