#### Recommended Citation

Postprint version. Published in *Europhysics Letters (EPL)*, Volume 64, Issue 4, November 1, 2004, pages 467-472.

The definitive version is available at https://doi.org/10.1209/epl/i2004-10234-2.

#### Abstract

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 **π**_{α} _{∈} Δ *C _{α}*-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

*C*1 and

*C*2 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).

#### Disciplines

Mathematics

#### Copyright

2004 EDP Sciences.

**URL:** http://digitalcommons.calpoly.edu/math_fac/35