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A proof of the associated sheaf theorem by means of categorical logic

  • Barbara Veit (a1)

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The double nature, both logical and geometrical, of topos theory is one of the most fascinating aspects of this discipline. Consequently, it might be of some interest that an essentially “geometric” fact such as the associated sheaf theorem admits a proof entirely based on methods of categorical logic.

The idea of this proof comes from a previous paper [V], where the same technique was used in the context of Grothendieck topoi. That paper used a generalized notion of forcing which leads directly from classical Tarski semantics in sets to Kripke-Joyal semantics in an arbitrary Grothendieck topos and gives a precise description of the links between the two. On account of this very close relationship, we thus could establish various basic facts on Grothendieck topoi without an extensive use of categorical methods, simply by viewing given subobjects as interpretations of appropriate formulas written in the formal language that had been used all along.

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[Bé]Bénabou, Jean, Structures syntaxiques, mimeographed notes by Roch Ouellet, Montreal, 1973.
[Bo]Boileau, André, Types vs. topos, thesis, University of Montreal, 1976.
[F]Freyd, Peter, Aspects of topoi, Bulletin of the Australian Mathematical Society, vol. 7 (1972), pp. 176.
[F.-M.]Farina, M. F. and Meloni, G. C., Il funtore fascio associato nei topoi, I. Una nuova costruzione, Pubblicazioni del dipartimento di matematica dell'Università di Calabria, Cosenza, 1977.
[G.-V.]Grothendieck, A. and Verdier, J. L., Théorie des topos (SGA 4), II.3.4., Lecture Notes in Mathematics, no. 269, Springer, New York, 1972.
[J 74]Johnstone, P. T., The associated sheaf functor in an elementary topos, Journal of Pure and Applied Algebra, vol. 4 (1974), pp. 231242.
[J 77]Johnstone, P. T., Topos theory, Academic Press, New York, 1977.
[L]Lawvere, F. W., Introduction to Toposes, algebraic geometry and logic, Lecture Notes in Mathematics, no. 445, Springer, New York, 1975.
[M.-R.]Makkai, M. and Reyes, G. E., First order categorical logic, Lecture Notes in Mathematics, no. 611, Springer, New York, 1977.
[T]Tierney, Myles, Axiomatic sheaf theory: some constructions and applications, Proceedings of the CIME Conference on Categories and Commutative Algebra, Varenna, 1971, Edizioni Cremonese, 1973.
[V]Veit, Barbara, Il forcing come principio logico per la costruzione dei fasci, I, II, Rendiconti di Matematica, Rome, 1978, vol. 11, Serie VI, fasc. 3, pp. 329353 and fasc. 4, pp. 601–625.

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