Olivia Caramello's website
Unifying theory
General introduction
Mathematics consists of several distinct areas: Algebra, Geometry, Analysis, Topology, Number Theory etc. Each of these areas has evolved throughout the years by developing its own ideas and techniques, and has reached by now a remarkable degree of specialization. With time, various connections between the areas have been discovered, leading in some cases to the creation of actual 'bridges' between different mathematical branches; think for example to analytic geometry, which allows to study geometrical shapes by using algebraic manipulation, or to algebraic geometry, algebraic topology, differential topology and differential geometry. In each of these cases, methods of one field have been employed to derive results in another, and this interplay of different points of view in a same subject has always had a fundamental role in illuminating the nature of concepts, establishing new results and suggesting new lines of investigation.
There are more and more instances of specialists in one field of Mathematics trying to find solutions to specific problems which apparently fall within the domain of their field, but instead fit much more naturally to different frameworks. It has happened several times that solutions to profound problems in one field have first, or only, been obtained by using methods from other fields, and this indicates that Mathematics should be seen as a coherent whole rather than as a collection of separate fields. To have an idea of this phenomenon, take for example analytic geometry; this theory provides a bridge between algebra and geometry, which can be fruitfully used to investigate problems both of algebraic and geometric nature. Indeed, it is often the case that certain algebraic properties of equations are best understood by using the geometrical intuition or conversely that certain geometrical properties are better investigated with the help of algebra.
In general, the importance of ‘bridges’ between different areas lies in the fact that they make it possible to transfer knowledge and methods between the areas, so that problems formulated in the language of one field can be tacked (and possibly solved) using techniques from a different field, and results in one area can be appropriately transferred through the 'bridge' to results in another field. This accounts for the central importance of investigations oriented towards the goal of unifying Mathematics, in the sense of finding large frameworks in which different mathematical phenomena can be interpreted as different instances of a unique abstract scheme, that is, as the result of applying a certain (small) number of abstract methodologies to different sets of concrete 'ingredients'.
By providing a system in which all the usual mathematical concepts can be expressed rigorously, Set Theory has represented the first serious attempt of Logic to unify Mathematics at least at the level of language. Later, Category Theory offered an alternative abstract language in which most of Mathematics can be formulated and, as such, has represented a further advancement towards the goal of 'unifying Mathematics'. Anyway, both these systems realize a unification which is still limited in scope, in the sense that, even though each of them provides a way of expressing and organizing Mathematics in one single language, they do not offer by themselves effective methods for an actual transfer of knowledge between distinct fields.
The kind of unification realized by these theories can be considered static, in the sense that it is achieved through a process of generalization, which allows to regard different concepts as particular cases of a more general one but does not offer by itself a way for transferring information between them:
For example, the fact that both preorders and groups are particular instances of the general notion of category does not give by itself a means for transferring results about preorders to results about groups or conversely.
On the other hand, the methodologies of topos-theoretic nature introduced in our work provide a systematic way for comparing distinct mathematical theories with each other and transferring knowledge between them. They are based on the use of abstract concepts called (Grothendieck) toposes as sorts of 'bridges' for transferring knowledge between different mathematical theories.
In fact, this latter methodology represents an instance of a different kind of unification, no longer based on the idea of generalization but rather on that of the construction of a 'bridge object' connecting to each other two given concepts.
This latter kind of unification, which we call dynamical unification (since it allows a 'dynamical' transfer of information between the two given objects), is characterized by the fact that two distinct objects are related to each other through a third one, which can be associated or constructed from each of them separately and which admits two different representations, each of which corresponding to a different method of constructing it. Such an object acts as a 'bridge' between the two given object in the sense that information can be transferred between the two objects by translating properties (resp. constructions) of the bridge object into properties of (resp. constructions on) the two objects, by exploiting the two different representations of the bridge object:
Of course, in general a given 'bridge' object may connect not only two objects with each other but many different pairs of objects. In fact, in the topos-theoretic setting, for each topos there exist infinitely many different mathematical theories associated to it (through the classifying topos construction).
In the case of toposes, the two objects to be related to each other are distinct mathematical theories which share a common 'semantical core', while the bridge object is a topos representing precisely this common 'core'.
Other instances of dynamical unification certainly occur in Mathematics; in fact, invariants are always sources of 'bridges' between objects on which they are defined (so, for example, the fundamental group of a topological space can be used a bridge for transferring information between topological spaces in the sense that if two topological spaces have isomorphic fundamental groups then certain topological properties, such as simple connectedness, can be transferred across the spaces; similarly, groups can be used to classify geometries etc.).
The startling aspect of toposes, as argued in this paper and more informally in this website, is that, unlike most of the invariants considered in Mathematics, they allow to compare with each other mathematical theories which can possibly belong to several different subfields of Mathematics, as well as to effectively transfer knowledge between them.