Olivia Caramello's website


Unifying theory


The following list of references, by no means complete, is meant to provide people interested in learning more about the unifying theory with some indications of materials which can be useful for acquiring the relevant technical background or deepening their understanding. 

Self-contained introductory text: click here for a draft of the first two chapters of my forthcoming book "Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'" (Oxford University Press, 2017). This 64 pages text is a self-contained introduction to toposes, categorical logic and the 'bridge' technique requiring only a basic familiarity with category theory. Comments are welcome!

The book "Topoi: The Categorial Analysis of Logic" by R. Goldblatt is a very gently-paced and self-contained introduction to Category Theory and the topos-theoretic approach to Logic, so it is particularly suitable for beginners who are not already familiar with the language of Category Theory.

As a first introduction to Topos Theory for people with a basic background in Category Theory, we recommend the book by S. Mac Lane and I. Moerdijk "Sheaves in Geometry and Logic".

The book "Sketches of an Elephant: a Topos Theory Compendium" in two volumes (the third volume of the trilogy is currenly being written) is the fundamental encyclopedic reference on Topos Theory. Even if this magnum opus is not intended as an introduction to Topos Theory and primarily aimed at specialists in the subject, some parts of it do not require an advanced background to be understood; in particular, we recommend Part D1-3 of the second volume as a self-contained introduction to Categorical Logic and the theory of classifying toposes. 

The fundamental work in which the concept of (Grothendieck) topos was originally introduced is "Théorie des topos et cohomologie étale des schémas (SGA 4)" by M. Artin, A. Grothendieck and J.-L. Verdier. This influential masterpiece is still very much cited today, and is a must-read for anyone seriously interested in Topos Theory and its relationships with Algebraic Geometry. 

A general description of the unifying methodologies is provided by this paper, while specific applications of these abstract techniques in different mathematical fields can be found in most of my papers (cf. this section for a list of notable examples and relevant references).

The book From a Geometrical Point of View by J.-P. Marquis is a philosophical study of the history and foundations of Category Theory, shedding light in particular on the nature of Topos Theory and its role within the mathematical sciences.

Finally, as a very general treatment of spectra (of first-order mathematical structures) from the perspective of classifying toposes, we recommend the paper "The Bicategory of Topoi, and Spectra" by J. C. Cole. The paper "Localisation, spectra and sheaf representation" by M. Coste provides more explicit descriptions of Cole's spectra and applications to sheaf representations of models of essentially algebraic theories.