Model Theory Inspired by Modern Algebraic Geometry: A Survey of Sheaf Representations for Categorical Model Theory

Gabriel Bittencourt Rios; Hugo Luiz Mariano

doi:10.14244/lajm.v2i01.8

Authors

Gabriel Bittencourt Rios University of Toulouse III
Hugo Luiz Mariano Universidade de São Paulo

DOI:

https://doi.org/10.14244/lajm.v2i01.8

Keywords:

Sheaves, Grothendieck Topoi, Categorical Logic, Schemes

Abstract

In this survey, we expound sheaf representations of categories in the context of categorical logic. Namely, we present classifying topoi of coherent theories in terms of equivariant sheaves of groupoid (and then explore the generalization of this technique to a more general categorical context); expose a representation of Grothendieck topoi as global sections of sheaf and, finally, show a quick introduction to logical schemes, a proposed model-theoretic analogue to the schemes of Algebraic Geometry.

Author Biography

Hugo Luiz Mariano, Universidade de São Paulo

Professor associado, Instituto de Matemática e Estatística

References

Awodey S, Forssell H, First-Order Logical Duality, Annals of Pure and Applied Logic, 2013 164 (3): 319-348. DOI: https://doi.org/10.1016/j.apal.2012.10.016

Awodey S, Logic in Topoi: Functorial Semantics for Higher-Order Logic, Dissertation, University of Chicago, 1997.

Awodey S, Sheaf Representations and Duality in Logic In: Casadio C (ed.), Scott P (ed.),Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics Springer International Publishing, 2021, pp. 39–57. DOI: https://doi.org/10.1007/978-3-030-66545-6_2

Breiner S, Scheme representation for first-order logic, PhD thesis, Carnegie Mellon University, 2014. Available at http://arxiv.org/abs/1402. 2600v1.

Bunge M, An application of descent to a classification theorem for toposes, Mathematical Proceedings of the Cambridge Philosophical Society , 1990, 107(1) :59–79. DOI: https://doi.org/10.1017/S0305004100068365

Butz C, Logical and Cohomological Aspects of the Space of Points of a Topos, PhD thesis, Utrecht University, 1996.

Butz C, Moerdijk I, Representing topoi by topological groupoids, Journal of Pure and Applied Algebra, 1998, 130(3): 223–235. DOI: https://doi.org/10.1016/S0022-4049(97)00107-2

Butz C, Moerdijk I, Topological Representation of Sheaf Cohomology of Sites,Compositio Mathematica, 1999, 118(2): 217-233. DOI: https://doi.org/10.1023/A:1001169215774

Caramello O, Theories, Sites, Toposes, Oxford University Press, 2018. DOI: https://doi.org/10.1093/oso/9780198758914.001.0001

Grothendieck A, Dieudonné J, Éléments de Géométrie Algébrique: Le Langage des schémas, Institut des Hautes Études Scientifiques, 1960. DOI: https://doi.org/10.1007/BF02684778

Johnstone P, Sketches of an Elephant: A Topos Theory Compendium, vol I, Oxford University Press, 2002.

Johnstone P, Sketches of an Elephant: A Topos Theory Compendium, vol II, Oxford University Press, 2002. DOI: https://doi.org/10.1093/oso/9780198515982.001.0001

Forssell H, First-Order Logical Duality, PhD thesis, Carnegie Mellon University, 2008.

Forssell H, Subgroupoids and Quotient Theories, preprint. Available at https://arxiv.org/abs/1111.2952.

Forssell H, Topological Representation of Geometric Theories, Mathematical Logic Quarterly, 2012, 58(6): pp. 380–393. DOI: https://doi.org/10.1002/malq.201100080

Funk J, Hofstra P, Steinberg B, Isotropy and crossed toposes, Theory and Applications of Categories, 2012, 26(24): 660 – 709.

Hodges W, A Shorter Model Theory, Cambridge University Press, 1997.

Johnstone P, Factorization theorems for geometric morphisms, I, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1981, 22(1): 3–17.

Joyal A, Tierney M, An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematician Society, 309, 1984. DOI: https://doi.org/10.1090/memo/0309

Makkai M, Reyes GE, First Order Categorical Logic, Lecture Notes in Mathematics, 611, Springer-Verlag, 1977. DOI: https://doi.org/10.1007/BFb0066201

Moerdijk I, The Classifying Topos of a Continuous Groupoid. I, Transactions of the American Mathematical Society, 1988, 310(2):629–668. DOI: https://doi.org/10.1090/S0002-9947-1988-0973173-9

Moerdijk I, The Classifying Topos of a Continuous Groupoid. II, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1990, 31(2):137–168.

Paré R, Schumacher D, Abstract families and the adjoint functor theorems In: Johnstone P (ed.), Paré R (ed.), Abstract families and the adjoint functor theorems, Lecture Notes in Mathematics, 1978, 661, pp. 1–125. DOI: https://doi.org/10.1007/BFb0061361

Mac Lane S, Moerdijk I, Sheaves in Geometry and Logic, Universitext, Springer-Verlag, 1994. DOI: https://doi.org/10.1007/978-1-4612-0927-0

Martins, Y. Cohesive 2-Topoi, October 2022.