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Unifying theory

Interdisciplinary applications and future directions

As it is clear from the above explanations, the scope of applicability of the unifying methodologies described above is extremely broad within Mathematics; in fact, it encompasses the whole of (finitary or infinitary geometric) first-order mathematics (due to the fact that every geometric theory admits a classifying topos) as well as many parts of second-order mathematics, such as for example general topology (due to the second-order nature of the concept of Grothendieck site). Higher-order logical theories are not presently tractable from a topos-theoretic viewpoint, since higher toposes (as presented for example in the book "Higher Topos Theory" by J. Lurie) are not yet understood from the point of view of the theories they classify, and their representation theory (providing a class of natural invariants and site characterizations for them) is not yet developed. Anyway, some significant steps have already been taken in this direction (cf. the ongoing work by M. Shulman on stack semantics and 2-categorical logic, as well as the project of Homotopy Type Theory), and upon a successful completion of these projects it will be possible to extend the realm of applicability of the unification methodologies to the context of higher-order logic, so to encompass most, if not all, of the theories considered in the current mathematical practice.

At the same time, these intradisciplinary methodologies could also be profitably applied in other subjects besides Mathematics. For example, in Physics they could be profitably used for analyzing and interpreting dualities, as well as for clarifying the relationship between quantum mechanics and relativity theory; in fact, physicists have already interpreted parts of quantum mechanics and relativity theory by using toposes of different kinds, and it would therefore be natural to explore the extent to which Topos Theory could act as a unifying framework between these two theories.

In Computer Science, toposes are already extensively used in the study of the syntax and semantics of programming languages; it is therefore natural to expect the unifying techniques to become effective tools in studying programs from a logical point of view as well as for comparing them with each other. On the other hand, these methodologies could also be implemented on a computer to obtain a sort of 'proof assistant' being able to generate non-trivial mathematical results in distinct fields.

Other natural contexts of application of these techniques include Music Theory (cf. for example the book "The Topos of Music" by G. Mazzola, representing an attempt to introducing a topos-theoretic analysis of composition and performance) and the foundations of Linguistics (e.g., investigation of the syntax and semantics of natural languages, comparative studies and the theory of translation).  

For a collection of metaphors and analogies between the view 'toposes as bridges' and concepts from other fields of knowledge, please refer to this section.