Olivia Caramello's website

 

•      Home     
•     Research    
•     Blog    
•     Unifying theory    
•     Papers    
•     Talks    
•     Teaching    
•     Videos    
•     CV    
•     Contact

Papers, books and monographs

Research 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 book

The following book, written jointly with Laurent Lafforgue, provides an introduction to Grothendieck topos theory:

“Sites et topos de Grothendieck : une introduction”,
611 pages, arXiv:math.CT/2508.21609 (2025)

Introductory texts

• This text, which is a preliminary version of the first two chapters of my book "Theories, Sites, Toposes: relating and studying mathematical theories through topos-theoretic 'bridges'", is a self-contained introduction to topos theory and the 'bridge' technique.

• On 14 December 2016 I successfully defended my HDR (mémoire pour l'obtention de l'habilitation à diriger des recherches) at the University of Paris 7. This document provides an introduction to my research work and a synthetic description of the contents of my main papers with an emphasis on their conceptual and methodological aspects; it presents in particular several selected applications of the 'bridge' technique in different mathematical fields obtained in the last years. The slides of my presentation (in French) are available here. The Jury's report ("rapport de soutenance") is available here.

• Here is another text, written with Laurent Lafforgue in French, providing an introduction to the theory of toposes as 'bridges'. This text, which is an expanded version of a recent talk given by Lafforgue at the Université de Nantes, is particularly suited for geometers who are not very familiar with categorical logic and the theory of classifying toposes.
  

• Here is a text providing a conceptual introduction to the the theory of topos-theoretic 'bridges', specifically written for the general public.

Research monographs

• O. Caramello and L. Lafforgue, Engendrement de topologies de Grothendieck, démontrabilité et opérations sur les sous-topos, arXiv:math.CT/2508.21134, 249 pages (2025)
  
  
• O. Caramello and R. Zanfa, Relative topos theory via stacks,
  arXiv:math.AG/2107.04417, 204 pages (2021)
  
  
• O. Caramello, Denseness conditions, morphisms and equivalences of toposes,
  arXiv:math.CT/1906.08737, 168 pages (2020)
  
  
• O. Caramello, A topos-theoretic approach to Stone-type dualities,
  arXiv:math.CT/1006.3930, 158 pages (2011)
  

Publications (in reverse chronological order) 

1. On fibred products of toposes (with L. Bartoli), arXiv:math.CT/2509.07719, 54 pages (2025)
2. Sheaves on bicategories (with E. Pivet), arXiv:math.CT/2507.20820, 73 pages (2025)
3. Morphisms and comorphisms of sites II -- Distributors of sites (with A. Osmond), arXiv:math.CT/2507.20932, 23 pages (2025)
4. Local fibrations and relative Diaconescu’s theorem (with L. Bartoli), arXiv:math.CT/2507.14379, 88 pages (2025)
5. De Morgan’s law in toposes I (with Y. Chamoun), arXiv:math.CT/2507.01869, 51 pages (2025)
6. Morphisms and comorphisms of sites I -- Double categories of sites (with A. Osmond), arXiv:math.CT/2505.08766, 38 pages (2025)
7. On a (terminally connected, pro-etale) factorization of geometric morphisms (with A. Osmond), arXiv:math.CT/2502.04213, 39 pages (2025)
8. The unifying notion of topos, published as a chapter in the book The Mathematical and Philosophical Legacy of Alexander Grothendieck, Chapman Mathematical Notes. Springer-Birkhäuser (Panza, M., Struppa, D.C., Szczeciniarz, JJ. eds., 2024), available here.
9. On morphisms of relative toposes (jointly with Léo Bartoli), arXiv:math/AG/2310.20691, 46 pages (2023)
10. Fibred sites and existential toposes, arXiv:math/AG/2212.11693, 45 pages (2022)
11. The over-topos at a model (jointly with Axel Osmond,
    arXiv:math.CT/2104.05650, 30 pages (2021)
12. La « notion unificatrice »  de topos, 38 pages, chapter in the book « Lectures Grothendieckiennes » (Spartacus IDH and SMF, 2022), available here
13. Grothendieck toposes as unifying ‘bridges’ : a mathematical morphogenesis, in the Springer book “Philosophy of Mathematics. Objects, Structures, and Logics”, available here 
14. On the dependent product in toposes (jointly with R. Zanfa), Mathematical Logic Quarterly 67 (3), 282-294 (2021), preprint version available as arXiv:math.CT/1908.08488
15. Some aspects of topological Galois theoryem> (jointly with L. Lafforgue), Journal of Geometry and Physics 142, 287-317 (2019).
16. Syntactic categories for Nori motives (jointly with L. Barbieri-Viale and L. Lafforgue), Selecta Mathematica 24(4), 3619-3648 (2018), preprint version available as arXiv:math.AG/1506.06113
17. 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 
18. Cyclic theories (jointly with N. Wentzlaff), Applied Categorical Structures 25 (1), 105–126 (2017), preprint version available as arXiv:math.CT/1406.5479 
19. 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 
20. Topological Galois Theory, Advances in Mathematics 291, 646–695 (2016), correction note here, preprint version available as arxiv:math.CT/1301.0300  
21. 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
22. 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   
23. 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 
24. Motivic toposes, arXiv:math.AG/1507.06271 (2015), 41 pages
25. General affine adjunctions, Nullstellensätze, and dualities (jointly with V. Marra and L. Spada), arXiv:math.CT/1412.8692, revised version currently in press in the Journal of Pure and Applied Algebra (pp. 1-34).
26. 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'"
27. Topologies for intermediate logics, Mathematical Logic Quarterly 60 (4-5), 335-347 (2014), preprint version available as arxiv:math.CT/1205.2547    
28. Gelfand spectra and Wallman compactifications, arxiv:math.CT/1204.3244 (2012), 50 pages
29. A general method for building reflections, Applied Categorical Structures 22 (1), 99–118 (2014), preprint version available as arXiv:math.CT/1112.3603 
30. Site characterizations for geometric invariants of toposes, Theory and Applications of Categories Vol. 26, No. 225, pp 710-728 (2012)
31. The unification of Mathematics via Topos Theory, arXiv:math.CT/1006.3930 (2010), 41 pages (Russian translation available as arXiv:math.CT/1104.0563 ),
    to appear in a Springer book of the series “Studies in Universal Logic” in 2020
32. 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
33. Syntactic characterizations of properties of classifying toposes, Theory and Applications of Categories Vol. 26, No. 6, pp 176-193 (2012)
34. 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
35. 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'"
36. Atomic toposes and countable categoricity, Applied Categorical Structures 20 (4), 379-391 (2012), preprint version available as arXiv:math.CT/0811.3547
37. 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
38. De Morgan classifying toposes, Advances in Mathematics 222 (6), 2117-2144 (2009), preprint version available as arXiv:math.CT/0808.1519
39. Fraïssé's construction from a topos-theoretic perspective, arXiv:math.CT/0805.2778 (2008), 17 pages, to appear in Logica Universalis
40. 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
41. A solution to recursive second-order linear equations,
    Quaderni del Dipartimento di Matematica dell’Università di Torino (12/2005), 10 pages
42.  An unconventional solution to an Eulerian problem,
    Quaderni del Dipartimento di Matematica dell’Università di Torino (12/2005), 14 pages