**bridges**'
for connecting different mathematical theories with each other and
allowing an effective transfer of knowledge between them.

**The unification of Mathematics via Topos Theory**".

**Grothendieck topos.
**The theory of Grothendieck toposes is written in categorical
language, but, unlike Category Theory, it is much more expressive, due to an additional degree of freedom implicit in the definition of topos. Indeed, a
**category** can be thought of as a pair of sets related by some structure satisfying certain properties; any set can be regarded as a category, but most of the categories arising in Mathematics are not of this form. In fact, the concept of category has, with respect to the notion of set, an additional degree of freedom.
**Toposes** are mathematical objects which are built from a pair, called a
**site**, consisting of a category and a generalized covering, called
**Grothendieck topology**, on it in a certain canonical way (the process which produces a topos from a given site can be described as a sort of
'completion'). Different coverings can be considered on a given category leading in general to inequivalent toposes; this gives the notion of topos one more degree of freedom with respect to that of category.
The existence of these three 'degrees of freedom' implicit in the concept of a topos (two for the notion of category and one for that of Grothendieck topology) can be exploited to build
'mathematical universes' in which mathematical theories find their natural home and can be effectively compared with each other.

**classifying topos** of the theory, which represents the natural framework in which the theory should be investigated, both in itself and in relationship to other theories. Two theories having the same classifying toposes (up to equivalence) are said to be
**Morita-equivalent**.

The basic idea underlying the unification methodologies is that the classifying topos
of a
theory can be effectively used as a '**bridge'** to transfer information between the theory
and any other theory
which is Morita-equivalent to it:

T

**automatized** in many cases. Indeed, using the methods of
Topos Theory one can obtain characterizations of the above kind for
several invariants, holding uniformly for any theory or at least for
wide classes of theories (and for certain classes of invariants such
characterizations can even be established in a purely mechanical way);
in presence of a Morita-equivalence, these characterizations will thus
be able to act as the **arches** of a bridge connecting the two theories, making it possible to transfer information between them.

The fundamental intuition underlying the unifying methodologies is that a given mathematical property can manifest itself in several different forms in the context of mathematical theories which have a common semantics but a different linguistic presentation.

A (Grothendieck) topos can be thought of as a mathematical object which condenses in itself the semantics of a mathematical theory, representing the body of properties of the theory which do not depend on its linguistic presentation but only on its semantics. To any mathematical theory (of a general specified form) one can associate a topos, canonically built from any presentation of it, which represents its
'semantical core'. Such an object is called the **classifying topos** of the theory.

Two mathematical theories have the same classifying topos (up to equivalence) if and only if they have the same
'semantical core', that is if and only if they are indistinguishable from a semantic point of view; such theories are said to be
**Morita-equivalent**. Conversely, every Grothendieck topos arises as the classifying topos of some theory.

Many important dualities and equivalences in Mathematics can be naturally interpreted in terms of equivalences between the classifying toposes of different theories; on the other hand, Topos Theory itself is a primary source of Morita-equivalences. It is fair to say that the notion of Morita-equivalence formalizes in many situations the feeling of 'looking at the same thing in different ways', and in fact it is ubiquitous in Mathematics.

Now, Grothendieck toposes can serve as 'bridges' for transferring
information between Morita-equivalent theories in the following sense. The existence of different presentations for the
'semantical core' of a given mathematical theory translates, at the technical level, into the existence of different representations (technically speaking,
**sites**) for the classifying topos of the theory.
**Topos-theoretic invariants**, that is properties of toposes or constructions on them which are invariant under equivalence, can then be used to transfer properties between the two representations of the classifying topos and hence between the two theories. The transfer of properties arises from the process of expressing properties of toposes in terms of properties of their representations: these relationships between properties of sites
and properties of toposes
are called
**site characterizations**.

Given a Morita-equivalence and a topos-theoretic invariant I, site characterizations for I play in this context the role of the 'arches' of the bridge, the 'deck' of the bridge being given by the equivalence of toposes expressing the Morita-equivalence. The topos-theoretic invariant I thus acts in this context as a tool enabling to bring certain information from one side of it to the other of the equivalence; different invariants allow us to transfer different information.

This methodology is very effective from a technical viewpoint due to the fact that the relationship between a site and the topos which it generates is technically very well-behaved (although highly non-trivial), to the extent that for many important classes of invariants there exist natural bijective site characterizations holding uniformly for any site, or at least for large classes of them. Also, there exist infinitely many topos-theoretic invariants (and one can always easily introduce new ones - since, by an obvious metatheorem, any property written in the language of Topos Theory is automatically invariant with respect to the notion of equivalence of toposes, namely categorical equivalence - and seek natural site characterizations for them).

We can thus think of a topos as an object which, together with all its different representations, embodies in itself a great amount of relationships existing between the different theories classified by it. Any topos-theoretic invariant thus behaves like a 'pair of glasses' which allows to discern certain information which is 'hidden' in a given Morita-equivalence.

These methods **unify** distinct mathematical theories with
each other in the sense that properties (resp. constructions) arising in
the context of theories which have a common 'semantical core' but a different 'linguistic presentation' come to be seen as
* different manifestations* of a

As a matter of fact, these methodologies define a
novel way of doing
Mathematics which is inherently '**upside-down'** with respect to the more traditional approaches in which one starts with simple ingredients and proceeds to combine them to build more complicated structures. Indeed, these methodologies take as primitive ingredients rich and sophisticated mathematical entities, namely
**Morita-equivalences** and **topos-theoretic invariants**,
and allow one to extract from them a huge amount of information relevant
for classical mathematics (it should however be noted that the fact that
these 'primitive ingredients' are intrinsically complex does not imply
that it is difficult to obtain such objects from the current
mathematical practice, as it can be seen from my papers).

This intrinsically **'top-down**'
approach is actually liable to generate a great number of surprising
insights into a variety of different mathematical contexts, a sort of
uniform '**rain** of results' into distinct subfields,
results which could
well be hard to establish, or even to merely imagine, by using the
specialized methods of the different fields. As in the case of rain it is
difficult, if not impossible, to predict where any single drop of it will
fall, so in the case of the unifying methodologies it is hard to predict
with certainty and exactness which results will be generated by the
abstract machinery; on the other hand, as the power of the rain lies in
the energy and matter which it *globally* generates, as well as
in its *uniform* nature, so the power of the unifying
methodologies lies in their 'global fruitfulness' and 'uniform'
ramifications spreading transversally across the various subfields
of Mathematics.

In fact, the **creative** power of
these methodologies, as well as their uniform and canonical nature,
makes them effective tools for assisting specialists of the various
domain in the discovery of specific results, as well as a concrete
**guide** for developing mathematics in a **cohesive**
and **conceptually inspired** way.

Incidentally, it should be noted that these
methodologies could also be generalized to the case of 'bridges' whose
deck is given by some kind of relationship between toposes which is not
necessarily an equivalence, in the presence of properties or
constructions of toposes which are invariant with respect to such a
relation. Nonetheless, the advantage of focusing on
Morita-equivalences is twofold; from a technical vewipoint, it is
convenient because *all* natural topos-theoretic properties that
one can formulate are automatically invariant with respect to
categorical equivalence of toposes, while from a conceptual viewpoint it
can be naturally interpreted as a unifying technique, since the properties of the
different theories come to be interpreted as different manifestations of
a *unique property *lying at the topos-theoretic level*.*

The unifying methodologies are also effective from a
**computational**
viewpoint. Indeed, the well-behaved representation theory of
Grothendieck toposes allows to translate properties and results between
different theories in a **uniform** and '**semi-automatic**
way'. In particular, one can generate a great number of non-trivial
insights in different mathematical fields without really making any real
creative effort. Of course, not all of the results generated in this way
are necessarily 'interesting mathematical theorems'; many of them can be
rather 'weird' according to usual mathematical standards, and in fact,
in order to get interesting results, one has to use one's mathematical
sensitivity to 'programme the machine' appropriately (that is, by
choosing topos-theoretic invariants and Morita-equivalences in a
sensible way); but these results
can still be quite deep, due to the non-trivial nature of the
representation theory of Grothendieck toposes.

Toposes can thus act as 'universal translators'
across different mathematical theories in a very effective and fruitful
way. From a technical point of view, the main reason for this
effectiveness is two-fold: on one hand, topos-theoretic invariants
usually manifest themselves in significantly different ways in the
context of different sites (think for example of the amalgamation
property on a category or the property of a topological space to be
extremally disconnected, which represent different manifestations of a
given topos-theoretic invariants, namely the property of a topos to
satisfy De Morgan's law, in the context of different sites - see
here for more details), while on the other hand the site
characterizations for them which formally express such manifestations
are essentially *canonical* and can be generally established in a
systematic, and in many cases even mechanical, way, due to the very
well-behaved nature of the representation theory of Grothendieck toposes
in terms of sites.

Concerning the level of 'mathematical
depth' of the translations established through the bridge technique, we
remark that it is clearly impossible to accurately predict it *a
priori*. Anyway, an upper bound for the complexity of such results
is given by the sum of the mathematical depth of the three parts of the
bridge, namely its 'deck' (given by the Morita-equivalence) and its two 'arches' (given by the site characterizations), which nonetheless can
be very considerable (due to the depth of topos-theoretic methods).

Actually, while in most cases the topos-theoretic
translations do not admit easy or natural elementary proofs, it may
happen that, after discovering a certain 'translation' through the
bridge method, one could find a 'concrete' proof of it which does not
involve toposes; but even in these cases one often realizes that, in
spite of its greater technical complexity, it is the topos-theoretic
proof that best illuminates the 'real reason' why the result is true,
and also that ultimately it would have been rather hard, if not
impossible, to arrive at the result or even imagine it without the
'guide' of toposes. Indeed, the topos-theoretic proofs arising from
applications of the bridge method are natural both from a technical
viewpoint (because of their *canonicity*) and from a conceptual
one (because of their *unifying nature*, which reveals itself in
the fact that the method relates to each other properties (or constructions)
which represent *different concrete manifestations* of a *
unique abstract property* (or construction) lying at a higher level.

In a sense, the 'upside-down' way in which the results are generated through the bridge method is so different from the traditional techniques that it defines by itself a new way of doing Mathematics, intrinsically intra-disciplinary in character, which owes its strength to the identification of 'centres of mathematical symmetry' from which mathematical reality can be naturally observed. In fact, the naturality of this point of view is witnessed by its technical fruitfulness and creative power, which reflects itself in the possibility of a 'canonical' and computationally effective generation of concrete applications in distinct mathematical domains.