Olivia Caramello's website


Unifying theory

This section of the website is devoted to an informal description of the intradisciplinary methodologies of topos-theoretic nature which I have eleborated since the beginning of my Ph.D. studies with the purpose of 'unifying' different mathematical theories with each other. For a more formal overview of these methods, as well as of the view on which they are based, please refer to this paper or to this more recent text.

General introduction

A non-technical introduction to the concept of unification and its different incarnations in Mathematics.

A brief historical digression

A brief excursus into the history of Categorical Logic, with a particular emphasis on the notion of classifying topos and the relationship between the concept of Grothendieck topos and that of elementary topos.

Toposes as 'bridges'

A non-technical explanation of the view 'toposes as bridges' on which the unifying theory is based.

Metaphors and analogies

A collection of metaphors and analogies for the non-specialist reader which can help illuminate the philosphical meaning of the unifying methodologies.

A more technical explanation

A brief technical description of the unifying methodologies in the language of Topos Theory.

Concrete examples

A selection of topos-theoretic 'bridges' belonging to different fields of Mathematics which I have established in the course of my research work.

Unification programme

A brief description of the research programme based on the unifying theory which I intend to pursue in the next years.

Interdisciplinary applications and future directions

A discussion on the level of generality and scope of applicability of the 'bridge-building' techniques, both in Mathematics and in other subjects, together with a set of indications of future research directions which could contribute to a further development of the unification programme.   

Questions and answers

My answers to the questions or comments on the unification programme that I most frequently receive from colleagues and interested people.


A few indications for further reading relevant to the unifying theory described in the sections above.

Controversy with category theorists

An explanation of the way in which the research programme of toposes as unifying 'bridges' has been received by the main experts in topos theory, of the initiative of clarification that I have undertaken and of its results.