By Berne B.J., Harp G.D.

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3 in [9]). This is how (the algebraic representation of) intuitionistic logic manifests itself in a topos. Another, closely related fact is that the subobjects of any object A in a topos form a Heyting algebra. 2. Topos theory and physics A large part of the work on topos theory in physics consists in showing how states, physical quantities and propositions about physical quantities can be represented within a suitable topos attached to the system [5, 6, 7, 8]. The choice of topos will depend on the theory type (classical, quantum or, in future developments, even something completely new).

Phys. 106, (1986) 321 [3] B. Schroer, Particle physics in the 60s and 70s and the legacy of contributions by J. A. 0371 [4] B. Schroer, Ann. Phys. 295, (1999) 190 [5] G. -J. Borchers, D. Buchholz and B. Schroer, Commun. Math. Phys. 219, (2001) 125, hep-th/0003243 [7] S. ˚ Aks; Journ. Math. Phys. 6 (1965) 516 [8] D. Buchholz and S. J. Summers, String– and Brane–Localized Causal Fields in a Strongly Nonlocal Model, arXiv:math-ph/0512060 [9] H. Grosse and G. Lechner, JETP 11, (2007) 021 [10] Works by Buchholz and Summers as well as by Grosse and Lechner, in preparation.

This point of view is expounded in detail in Bell’s book [1], which is our standard reference on these matters. Other excellent sources are [24] and part D of [21]. The basic concept consists in deﬁning a formal language and then ﬁnding a representation of it in a suitable topos. As usual in mathematical logic, the formal language encodes the syntactic aspects of the theory and the representation provides the semantics. Topoi are a natural ‘home’ for the representation of formal languages encoding intuitionistic logic, more precisely, intuitionistic, higher-order, typed predicate logic with equality.