Olivia Caramello's website

 

Papers

Book

My book "Theories, Sites, Toposes: relating and studying mathematical theories through topos-theoretic 'bridges'" has recently been published by Oxford University Press. It can be bought for instance through Amazon or the OUP website.

Here is the back cover:

This book introduces a set of methods and techniques for studying mathematical theories and relating them to each other through the use of Grothendieck toposes. The theory of classifying toposes – which geometrically embodies the mathematical content of first-order (geometric) theories - is first recalled, allowing the formulation of general ‘bridge’ principles: study theories through the computation of invariants of their associated toposes in terms of different presentations of these toposes. As any Grothendieck topos has infinitely many presentations, the expression of its invariants in terms of them gives rise to a veritable mathematical morphogenesis. These methods, which are susceptible to unify notions and results across distinct mathematical areas, are applied in particular to the study of geometric theories and their extensions through suitable topos-theoretic invariants. The book concludes with a selection of applications of the theoretical results obtained in the previous parts to very different concrete mathematical theories.

Introductory texts

Publications (in reverse chronological order)

  1. Sur la dualité des topos et de leurs présentations et ses applications : une introduction (in French, jointly with L. Lafforgue), available here (2016), 61 pages
  2. On the geometric theory of local MV-algebras (jointly with A. C. Russo), Journal of Algebra 479, 263-313 (2017), preprint version available as arXiv:math.CT/1602.03867 
  3. Motivic toposes, arXiv:math.AG/1507.06271 (2015), 41 pages
  4. Syntactic categories for Nori motives (jointly with L. Barbieri-Viale and L. Lafforgue), arXiv:math.AG/1506.06113 (2015), 25 pages
  5. General affine adjunctions, Nullstellensätze, and dualities (jointly with V. Marra and L. Spada), arXiv:math.CT/1412.8692 (2014), 43 pages
  6. Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective (jointly with A. C. Russo), Bulletin of Symbolic Logic 22 (2), 170-214 (2016), preprint version available as arXiv:math.CT/1409.4730 
  7. Cyclic theories (jointly with N. Wentzlaff), Applied Categorical Structures 25 (1), 105–126 (2017), preprint version available as arXiv:math.CT/1406.5479 
  8. Extensions of flat functors and theories of presheaf type, arxiv:math.CT/1404.4610 (2014), 158 pages, incorporated in the book "Theories, Sites, Toposes: relating and studying mathematical theories through topos-theoretic 'bridges'"
  9. The Morita-equivalence between MV-algebras and abelian l-groups with strong unit (jointly with A. C. Russo), Journal of Algebra 422, 752–787 (2015), preprint version available as arxiv:math.CT/1312.1272 
  10. Topological Galois Theory, Advances in Mathematics 291, 646–695 (2016), correction note here, preprint version available as arxiv:math.CT/1301.0300    
  11. Topologies for intermediate logics, Mathematical Logic Quarterly 60 (4-5), 335-347 (2014), preprint version available as arxiv:math.CT/1205.2547    
  12. Gelfand spectra and Wallman compactifications, arxiv:math.CT/1204.3244 (2012), 50 pages
  13. Priestley-type dualities for partially ordered structures, Annals of Pure and Applied Logic 167 (9), 820-849 (2016), correction note here, preprint version available as arXiv:math.CT/1203.2800   
  14. A general method for building reflections, Applied Categorical Structures 22 (1), 99–118 (2014), preprint version available as arXiv:math.CT/1112.3603 
  15. Site characterizations for geometric invariants of toposes, Theory and Applications of Categories Vol. 26, No. 225, pp 710-728 (2012)
  16. A topos-theoretic approach to Stone-type dualities, arXiv:math.CT/1006.3930 (2011), 158 pages
  17. The unification of Mathematics via Topos Theory, arXiv:math.CT/1006.3930 (2010), 41 pages - Russian translation available as arXiv:math.CT/1104.0563
  18. A characterization theorem for geometric logic, Annals of Pure and Applied Logic 162 (4), 318-321 (2011), preprint version available as arXiv:math.CT/0912.1404
  19. Syntactic characterizations of properties of classifying toposes, Theory and Applications of Categories Vol. 26, No. 6, pp 176-193 (2012)
  20. Universal models and definability, Mathematical Proceedings of the Cambridge Philosophical Society 152 (2) 279-302 (2012), preprint version available as arXiv:math.CT/0906.3061
  21. Lattices of theories, arXiv:math.CT/0905.0299 (2009), 84 pages, incorporated in the book "Theories, Sites, Toposes: relating and studying mathematical theories through topos-theoretic 'bridges'"
  22. Atomic toposes and countable categoricity, Applied Categorical Structures 20 (4), 379-391 (2012), preprint version available as arXiv:math.CT/0811.3547
  23. De Morgan's law and the theory of fields (jointly with P.T. Johnstone), Advances in Mathematics 222 (6), 2145-2152 (2009), preprint version available as arXiv:math.CT/0808.1972
  24. De Morgan classifying toposes, Advances in Mathematics 222 (6), 2117-2144 (2009), preprint version available as arXiv:math.CT/0808.1519
  25. Fraïssé's construction from a topos-theoretic perspective, arXiv:math.CT/0805.2778 (2008), 17 pages, to appear in Logica Universalis
  26. Yoneda representations of flat functors and classifying toposes,
    Theory and Applications of Categories, Vol. 26, No. 21, pp. 538-553 (2012)

    My Ph.D. thesis is available upon request.

    Previous work

    The following papers were written during my undergraduate studies.
    fg
  27. A solution to recursive second-order linear equations,
    Quaderni del Dipartimento di Matematica dell’Università di Torino (12/2005), 10 pages
  28.  An unconventional solution to an Eulerian problem,
    Quaderni del Dipartimento di Matematica dell’Università di Torino (12/2005), 14 pages