Abstract
For an arbitrary category \(\mathcal{C}\), the category \(Sets^{\mathcal{C}^{op}}\) of presheaves is actually a topos. This topos is important since, for a particular choice of \(\mathcal{C}\), it will be the topos we will utilise to express quantum theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that the last product is all in Sets
- 2.
A category \(\mathcal{C}\) is said to be locally small iff every hom-set \(\mathcal{C}(A,B)\) forms a proper set.
- 3.
We have already encounter this in Example 5.10 of Chap. 5.
- 4.
Recall from Chap. 4 that the comma category \(\mathcal{C}\downarrow a\) has, as objects, all maps in \(\mathcal{C}\) with codomain a, i.e. all maps of the form f:b→a. Given two objects f:b→a and g:c→a a morphism between them is a \(\mathcal{C}\) arrow h:b→c such that g∘h=f.
- 5.
Here the contravariant functor G a is defined in an analogous way as F a .
References
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic: AÂ First Introduction to Topos Theory (Springer, London, 1968)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Flori, C. (2013). Topos of Presheaves. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-35713-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35712-1
Online ISBN: 978-3-642-35713-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)