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.