*Europhys. Lett.*,

**68**(4), pp. 467-472 (2004)

DOI: 10.1209/epl/i2004-10234-2

## Elemental principles of t-topos

**G. Kato**

Mathematics Department, California Polytechnic State University San Luis Obispo, CA 93407, USA

gkato@calpoly.edu

received 3 February 2004; accepted in final form 20 September 2004

published online 29 October 2004

** 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 by , *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
is defined as the category of
-valued presheaves on the
*t*-*site*
*S*, where
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).

**PACS**

03.65.Ud - Entanglement and quantum nonlocality (

*e.g.*EPR paradox, Bell's inequalities, GHZ states, etc.).

03.65.Ta - Foundations of quantum mechanics; measurement theory.

04.20.Cv - Fundamental problems and general formalism.

**©**

*EDP Sciences 2004*