Topos


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In this chapter, with the aid of some examples, I will describe what a Topos is. In particular I will focus on the Topos of Presheaves, since it will be used in section Topos and Logic of this websit, to describe the new type of "Quantum Logic" put forward by Isham and Butterfield [1], [6], [8], [2], [4].

A Topos, generally speaking is a category in which a number of basic constructions of a category are always possible.
The formal definition goes as follows

Definition 1.1   A Topos is a category T with the following extra properties:
  • T has an initial (0) 1.1 and a terminal (1) object 1.2


  • T has pullbacks 1.3


  • T has pushouts 1.4


  • T has exponentiation 1.5 i.e. T is such that for every pair of objects X and Y in T exists the map $ Y^X$


  • T has a subobject classifier 1.7

More information on the detailed definition of a Topos can be found in [9], [11].


Subsections
Cecilia Flori 2007-01-02