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.

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.

Resources

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