Abstract
In this chapter we will explain how to define categorical notions internally within a topos. This internal description of objects is needed to understand the covariant approach to topos quantum theory explained in the next chapter.
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Notes
- 1.
KRegLoc represents the category of compact regular locales. A locale L is compact if every subset S ⊆ L with \(\bigvee S =\top \) has a finite subset F with \(\bigvee F=\top \). It is regular if every element of L is the join of the elements well inside itself. Given two elements, a, b then a is well inside b (denoted a ≪ b) if there exists c with c ∧ a = ⊥ and c ∨ b = ⊤.
- 2.
It is a standard result that in every topos the sub-object classifier is a Heyting algebra.
- 3.
In the case at hand it means that h ∗ commutes with arbitrary unions and finite intersections, respectively.
- 4.
A bicartesian closed poset is a poset which (when thought of as a thin category) is (a) finitely complete, (b) finitely cocomplete and cartesian closed.
References
B. Banaschewski, C.J. Mulvey, The spectral theory of commutative C*-algebras: the constructive Gelfand-Mazur theorem. Quaest. Math. 23(4), 465–488 (2000)
B. Banaschewski, C.J. Mulvey, A globalisation of the Gelfand duality theorem. Ann. Pure Appl. Logic 137, 62–103 (2006)
P.T. Johnstone, Sketches of an Elephant A Topos Theory Compendium I, II (Oxford Science Publications, Oxford, 2002)
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, London, 1968)
S.A.M. Wolters, Quantum Toposophy UB Nijmegen [host] (2013)
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Flori, C. (2018). Internalizing Objects in Topos Theory. In: A Second Course in Topos Quantum Theory. Lecture Notes in Physics, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-319-71108-9_9
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