 Today I'm going to present to you some ideas around unification and morphogenesis, which were inspired by my work on using Grotten diktoposis as unifying bridges across different mathematical contexts. So the structure of the talk is like this. So I will start by talking about the general concept of unification in mathematics by distinguishing between two distinct senses in which the term unification can be taken. In particular with the purpose of clarifying the sense in which we talk about unifying bridges. Then I shall proceed to review in a conceptual non-technical way this theory of toposis as bridges that I have been developing throughout the past years. And then I will come to the main message of the talk, which is explaining the sense in which the duality between topospheric invariants and their manifestations represent a sort of mathematical morphogenesis. Then I shall present some more general reflections motivated by these technical developments in mathematics and toposphere. More precisely, I shall present the general idea of a bridge object, an idea which actually makes sense well beyond the mathematics as we shall see instances of bridges can be found in essentially any field of knowledge. And then there are also other aspects of the theory of toposis as bridges which can be abstracted and which makes sense in a more general context as well. In particular, we shall talk about the theme of symmetries and the strict relationship that this theme entertains with that of completion processes for objects that one wants to unify with other objects. And then if some time remains, I shall sketch some future directions that these studies paved the way for. Okay, so let's then start talking about unification and bridges in mathematics as they are normally conceived. So, mathematics traditionally is divided into several distinct areas, algebra, geometry, analysis, topology, numbers, theory, etc. And each of these areas, which by now have reached a remarkable degree of specialization, is characterized by its own set of languages and techniques. Still, these different areas should not be thought as separate compartments that have no connections with each other. In fact, since the very first historical developments of mathematics, various connections between different mathematical areas have been discovered, leading in some situations to the creation of actual bridges enabling an effective transfer of information across different mathematical fields. The most elementary but non trivial example of bridge with whom I think anyone having attended high school is familiar with is analytic geometry, which allows to use algebraic manipulation to think about the basic entities of Euclidean geometry and conversely, it provides a geometric intuition on some simple kind of equations. Now, in general, bridges are very important because, as I already said, they make it possible to transfer knowledge across different areas, and not only they allow one to dynamically investigate mathematical context because they also allow to multiply points of view on a given theme. So, one should not just think of bridges as necessarily connecting two very different entities with each other, but as we shall see, bridges can also be used for investigating a single mathematical theory, but in a very dynamical way. Now, what I would like to start explaining is the role of mathematical logic and the topos theory, which in fact are also quite related to each other, in relation to the goal of investigating in a fully systematic and rigorous way the relations existing between different mathematical theories because it is natural to wonder can we build a sort of theory of bridges so that we can say classify bridges with any effect logic and topos theory really provide a very effective tools in relation to this goal as we shall explain in a minute. But before then, I would like to clarify what is usually meant by unification because the term unification by itself is a bit ambiguous. So it's important to be precise about the precise meaning of the term we are going to use in relation with bridges. So normally when one thinks about a unifying framework, one has the idea of like a global broad framework, like a very big box into which several different, say particular instances can be fit into. So one has this idea in particular when you think about set theory as a unifying language for mathematics or category theory, etc. One has the idea of unification, which is like a broad framework and compassing a high number of particular cases. So, so this certainly is a possible meaning to the word unification, but it is what I call here a static unification, which is achieved by means of a generalization process. Why do I call it static? Well, because when you generalize, for instance, two concepts to a more general one, this does not provide by itself a means for transferring information between the two objects that you are unifying. Because actually, what you can say is that whatever applies in general to the to the more general concept applies as the arrows indicate to each of the two particular instances. But if you start from one particular instance, and you want to transfer some information to the other, you don't have an effective way to proceed. And actually, when when you deal with this broad global frameworks, you are really dealing with some broad kind of generalization and so you are speaking of unification in this first sense. But there is another meaning to the word unification, which is much more interesting and substantial, and it is precisely in this sense that our bridges are going to be unifying. It is what I call dynamic unification. So the process by by by means of which such such a kind of unification occurs is not through a generalization, as in the first case, but it's through a sort of construction. So by construction, I mean any way of associating, not necessarily in a concrete sense, but even abstractly, a third object to each one of a pair of two objects that one wants to relate with each other. So the idea is that in order to put two objects in in connection with each other, it might be good to introduce a sort of third point of view on the two objects provided by some other kind of object that could act as a bridge enabling transfers of information between them. So this is a very vague intuition. I'm going to be more precise about this later in the talk, but I wanted just to introduce the vague idea. So the idea is that if one is able to associate, say the same kind of object, the same object to each of the two objects that one wants to relate, then what is possible to do is to consider properties of this bridge objects from the two perspectives from the perspective of the first object, which will yield a certain representation of the bridge object. And on the other hand, in terms of the second object, which will also provide another alternative representation for the bridge object. And in this way, one will establish connections, which indeed as the arrows indicate will actually allow a certain kind of transfer of information across the two objects. So if we revisit the frameworks we have mentioned from the point of view of this two kinds of unification, we realize that set theory and category theory realize actually a static unification of mathematics, essentially of linguistic nature, because in fact they provide an abstract language in which, if you want, you can say codify, you can express most of mathematics. Of course, the language of set theory is different from that of category theory, but both of them are extremely expressive in terms of capability of capturing a great amount of different mathematical entities. But you see that, in fact, providing a way of expressing and organizing mathematics in one single language does not provide by itself effective methods for performing an actual transfer of knowledge between different fields of mathematics, for instance. So one needs indeed to resort to the other kind of unification, which is the dynamic form of unification mentioned earlier. Indeed, toposes as spaces on which a number of fundamental mathematical invariants are naturally defined, allow indeed to act as bridge objects for connecting with each other a great deal of different mathematical theories. In fact, more generally, whenever you have a significant invariant in mathematics, what is going on is really a dynamic form of unification. So toposes are particularly powerful in serving as unifying bridges because they support a great number of significant invariants. So in fact, most of the invariants that are crucial in mathematics think, for instance, of homological invariants or homotopy theoretic invariants, logical invariants, etc. They are actually invariants of toposes. So this is the way you should think about the toposes in the context of this seminar. I'm not going to use the technical definition of topos because of course it would require several lectures just to give such a definition and we shall not really need this definition for the general ideas I'm going to present in this talk. But if you just want to get an idea of the topos, you can really think of it as a universe which supports in a very natural way a great amount of significant mathematical invariants. And indeed, toposes, because of this have an incredible unifying potential, which was already glimpsed by Grotendick, the inventor of the concept of topos. Here is a very striking excerpt from Rekolt Semi, this wonderful autobiographical text by Grotendick, where he comments on his mathematical discoveries also from a very enlightening conceptual viewpoint. So here as you can see what Grotendick says is that toposes actually can be meaningfully associated with many different mathematical contexts in a significant way. You see the word that Grotendick uses is unisense commune, so a common essence of mathematical situations that can be very distant from each other and coming from one region or another of the vast universe of mathematical things. So indeed, it's like this, so toposes can capture by in a significantly invariant way a great number of the essence of a great number of different mathematical situations. So Grotendick remarked that indeed toposes are unifying in the sense that you can attach, say the same kind of object to situations that can come from different fields of mathematics such as geometry, algebra, topology, arithmetic, mathematical logic in category theory. But it didn't go further than that in the sense that it didn't talk about how toposes could actually be used for, say, building bridges across these areas in the sense of transferring knowledge between these areas. And in fact, this has been my main concern since the beginning of my PhD studies. So my aim has always been that of developing a number of techniques, allowing one to indeed exploit this great unifying potential of the notion of topos to establish surprising and possibly very deep connections across the different areas. So in 2010, I wrote a programmatic text containing a number of general principles, like suggesting ideas on how to use the technical flexibility inherent to the concept of topos to actually build such bridges. In fact, the main principle introduced in that paper on which all the theory revolves is the principle of presenting a topos in a multitude of different ways and of considering invariance of toposes from the point of view of these different presentations. So this is the key idea. And so since I had this idea, I worked very, very hard to test its validity in different areas to see whether it made sense, whether it was a powerful tool of discovery. And indeed, since the very beginning, this principle turned out to be very fruitful. It allowed to obtain solutions to a great number of longstanding problems in categorical logic, but most importantly, it led to several substantial applications in different mathematical fields. And not only it paved the way for a new way of investigating mathematical theories in a very dynamical way, because as I already mentioned, these bridges are not just useful for connecting with each other theories that belong to different areas, but they can also provide effective tools to study just one single theory by a multiplicity of points of view, and so in a very dynamical way. Still, we are at the very beginning of explorations in this direction. Of course, these are very interdisciplinary studies. So as you can imagine, much remains to be done both theoretically, because also toposphere should be further developed also on the theoretical level, but also from the point of view of applications, there is a lot to do in order to make this technique more and more user friendly. And so actually used by the working mathematician and to actually realize the goal of making tools, making these fundamental tools for investigating mathematical theories and their relations. So here is just a list of some applications of topospheric bridges obtained so far. In fact, if you wanted to have an overview of such applications as well as many others, you can refer to my habilitation thesis, which is the 100 pages document, which should be readable by a mathematician who is not necessarily a specialist in category, purely logical or toposis. So you will see that actually what is going on is really like the general abstract theory, but applied in a great variety of different situations and leading to insights that are actually quite deep in each of these areas. So I find this quite striking and illustrative of the unifying potential of the notion of topos already observed by Grotendick. Okay, so now I wanted to go a bit more in detail about this use of toposis as unifying bridges and to tell you in particular about the connection between toposis and logic. So at the beginning of the talk, I remarked that both mathematical logic and topos theory could be fundamental tools for studying mathematical theories and their relationships. And indeed, the two subjects are quite related to each other and the perspective that actually establishes this connection is that of toposis as the classifiers of suitable kinds of structures that can be formalized within a certain kind of logic. So this perspective was originally introduced by Grotendick in some particular cases in the context of algebraic geometry, and then in the 70s a number of categorical logicians and most notably the Montreal School of Categorical Logic, led by Mackay and Reyes, introduced the first logical framework, namely geometric logic, in which every theory that is formulated admits a classifying topos. So I'm not going to give any details of what the geometric logic is, but let me just say that it's a very expressive kind of logic which goes even beyond to a certain extent finitary first order logic. Because it also has an infinitary character. So I mean, when you talk about a theory in this framework, well, you can think about a theory belonging to possibly any area of mathematics. So, as you know, first order theories and also geometric theories can be found in any area of mathematics. In fact, classical finitary first order theories are the object of study of the field of mathematical logic, which is model theory, as you know. And here what is going on is actually a factorial model theory, because, as the name suggests it, classifying toposes are toposes which classify the models of mathematical theories, not just the classical set base models, but the models in arbitrary toposes. So in fact, such a classification, it's important to remark that doesn't exist in the restricted context of set theory. But if you consider models of mathematical theories, not just in sets, but in arbitrary toposes, then you get the existence of a particular topos associated with the theory, called this classifying topos, which will contain a particular model of the theory called its universal model. And indeed this model will classify all the others by sort of deformations induced by certain kinds of structure preserving functions. So you have a very, very beautiful classification result at the cost of enlarging your view. In fact, this is a theme on which we shall come back later in the seminar, the idea of enlarging the view in order to get the symmetry. And this is one illustration of this remark on which we shall say more later on. Okay, so now that we have talked about the existence of classifying toposes for a wider class of theories, we can wonder whether it is possible for two theories to have the same classifying topos. Yes, indeed it is possible and when it happens, we say that the theories are more equivalent, but when does it happen exactly? Well, by definition of classifying topos, two theories have the same classifying topos if and only if they have equivalent categories of models in any topos, naturally in the topos. So it means that essentially the two theories describe the same semantics, the same mathematical content, the same structures in different languages. So what is going on here is really like a bridge phenomenon because we have like if two theories are more equivalent, they can be connected by the common classifying topos. In fact, any topos is actually the classifying topos of some theory and in fact of infinitely many theories. This is a theory. And therefore we can conclude that every topos can be seen as a canonical representative for equivalence classes of theories modulo this equivalence relation called orita equivalence. So, in fact, the notion of Morita equivalence is extremely deep and interesting because indeed it formalizes in many situations the feeling of looking at the same thing in different ways or constructing a given object through different methods. And indeed it is ubiquitous in mathematics. In fact, one is if one doesn't work in topos theory one does not really look for such Morita equivalences and so one doesn't have the perception that there are Morita equivalences everywhere. But indeed when you studied the subject in a systematic way you discovered that bridges are everywhere Morita equivalences are everywhere. And so it's not like a miracle when a bridge happens. In fact, it's more the rule than the exception. But of course you need to have a framework such as that of topos theory in order to really see all these connections in a natural way. In fact, you might wonder if you already have some dualities or equivalences in mathematics whether you can interpret them in terms of Morita equivalences and indeed this is the case. I mean in my work I have analyzed a great number of known classical dualities for instance the stone type dualities or dualities arising in the theory of lattice ordered groups and the algebra etc. So many important dualities indeed can be naturally understood from this point of view and whenever you understand them from a topos theoretic viewpoint this gives you further information. It allows you to get further insights and to understand much better what is going on. In fact, there is not a one-to-one correspondence between Morita equivalences and dualities or equivalences. For instance, stone type dualities are obtained by functorializing a bunch of bridges. So, for instance, if you want to get stone duality you don't just need a bridge, you have one bridge for each pair of structures that are related by the duality. And by functorializing all these bridges, this is how you get the duality. So, in fact, the relationship between dualities, correspondences, equivalences and Morita equivalences is by no means straightforward but still it is quite natural. Then, of course, one might wonder what happens if you have a sort of dictionary between two theories, which technically speaking is called a by-interpretation in logic. So, of course, when two theories are by-interpretable then, of course, they are semantically equivalent and so they are necessarily Morita equivalent. But what is very interesting is that the converse doesn't hold. And in fact, you can prove a sort of metaphor saying that the majority of Morita equivalences which arise in mathematics do not come from dictionaries. So this means that if you reason concretely you will miss most of them. So this means also that most of the connections arising in mathematics are actually hidden from the naked high. So we need some technology in order to discover them. So this is quite important. Then, as I already said a couple of times, bridges are useful not just for relating distinct theories with each other but also for investigating a single theory in a dynamical way. In fact, when you give the axioms of a theory, it's like giving birth to a living organism. And so the theory starts developing and so new results are discovered in the theory. And so you have a whole dynamics of proofs in a theory, etc. And this, of course, provides different points of view on your theory. For instance, suppose you have the theory of fields. You can regard the theory of fields as an extension of the theory of rings, or you can regard that as an extension of the theory of integral domains, etc. These are quite the trivial observations, but from a technical viewpoint, they give rise to different presentations for the classifying topos of that theory. Which can really also have technical consequences because when you want to compute certain invariance, then some representation could be better than another. And so even some very apparently trivial remarks from a concrete viewpoint could lead to significant insights when you insert these ingredients into the general topospheric machinery. Now, the importance of Morite equivalence lies also in the fact that it can be very well understood by means of the methods of topos theory. Indeed, different presentations of the same topos can be interpreted as Morite equivalences between different theories. And conversely, the existence of different theories with the same classifying topos translates at the technical level into the existence of different presentations for the same topos. So basically having different theories classified by the same topos is the same thing as having different ways of presenting a topos. So topos is traditionally represented by using categorical concepts that are called sites. So a site is a pair consisting of a category, usually a small category, equipped with a notion of covering of objects in the category by families of arrows going to it. Which is called a groten-dictopology. So a site is a pair consisting of these two things, a category and a groten-dictopology on it. In any case, topos can also be presented by using other kinds of objects like group oids, quantals, etc. But sites are, say, the most common way of presenting topos, and indeed they are also very well behaved computationally. I mean, when you want to compute invariance, in fact, invariance tends to admit quite pleasant site characterizations. It's also true for other objects presenting topos, but for sites it is certainly true. And so basically having different theories classified by the same topos, technically speaking, translates into the existence, for instance, of different sites which present the same topos. A topos is, by definition, a category equivalent to the category of ships on some site. So from a site you can canonically build a topos by taking all the ships on the site. Okay, in any case, still using this logical perspective of classifying topos is the bridge technique works like this. Suppose you have two theories, T and T prime, which are more equivalent, namely that they had equivalent classifying topos. You can consider topos theoretic invariance on the common classifying topos. By invariant I mean whatever property or construction on toposes, which can be transferred across categorical equivalents of toposes. Let me just say that it is not difficult to find topos theoretic invariance because essentially whatever natural property you write in categorical language, will be more or less automatically invariant with respect to categorical equivalents. Because the criterion of identity for toposes is simply you take the underlying categories and you see whether they are equivalent. So it is a simple notion of equivalence and so it's easy to see to detect whether a certain property is invariant or not. So this means that we dispose of a really great infinite number of invariants. And for each of these invariants, we can look at how this invariant expresses on the one hand in terms of one theory and on the other hand in terms of the other. So suppose for instance you have one theory T, which is of algebraic nature, another theory T prime, which is of geometric nature and suppose they have the same classifying topos when you play this game. When you unravel a certain invariant in terms of the algebraic theory, you will get an algebraic property. When you do this unraveling in terms of the other theory, you will get a geometric property and these properties will be equivalent to each other just because they are interpreted as different manifestations of a single invariant lying at the topos level. So this is how the transfer of information takes place through the bridge technique. So let me just stress that you can actually any invariant induces a bridge. So there is not just one bridge connecting different theories but there are infinitely many bridges because each invariant induces a bridge. And in fact, each invariant allows to transfer a certain kind of information. So what is going on here is really a form of unification because, as I said, different properties arising in the context of different theories come up as different manifestations of a unique property lying at the topos theoretic level. So we have a unity lying at the topos level and a diversity lying at the level of theories or sites or presentations of toposes. And indeed, you cannot reduce to having a single bridge. So because there is not one invariant that is more important than all the others and that subsumes all the others. Each invariant which allows you to transfer different information. And so if you want to collect a lot of information about the context that you want to investigate, it's convenient to use a number of selected invariants. So in my papers, normally to get a good number of insights, I tend to use say 10, 15 invariants or something like that. So if you use like an even a limited number of invariants, you can still transfer interesting information and get new insights on theories. I shall give some examples later. So why is this methodology technically feasible? Well, actually, to understand this, of course, you would need to be a specialist in topos theory. But let me just say that, in fact, the reason is that the structural relationship which exists between a topos and its presentations is something very natural. Very natural in the sense that it allows to perform the some ravelings of invariants quite easily. Of course, it depends on the kind of invariants. There are some invariants, for instance, the homological ones that are actually very hard to compute in terms of sites or other presentations for toposism, even in particular cases. But on the other hand, there are why the classes of invariants which admit even automatic unravelings in terms of different presentations. So we have a great variety of complexity for the some ravelings of invariants. And of course, this accounts for a different degree of mathematical depth for the results generated through the bridge technique. So indeed, I mean, what I can say is that even when you use invariants which admit automatic reformulations in terms of different presentations and there are many of them, you can get very surprising and deeper results by using the bridge technique. Of course, when you use invariants whose computation is very sophisticated, then the complexity of the translation that you perform is even higher. But still, I mean, this is definitely a non-trivial technique even when applied to invariants that can be easily computed. Okay, so here is a more technical picture depicting a bridge such as the ones that you can find, for instance, in my book. So in fact, here I have used the sites for presenting toposis. So you see two different sites, CJ on the one hand and DK on the other, which presented the same topos. So the topos of shifts on the first site is equivalent to the topos of shifts on the second. And in such a situation, as I said, what you can do is for you take whatever invariant I and you look at this invariant from the point of view of the first site. And then from the point of view of the second. So if you are able to effectively perform these unravelings, you will end up with a property P CJ of the first site and the property Q DK of the second site. And in fact, these properties will be equivalent because they correspond to the same abstract property lying at the topos theoretically then. So what is interesting about this is that I mean explain like this it seems like a total logical, but in practice it is not because if these unravelings are natural enough. These properties, these concrete properties on one side and on the other they can be completely different. And so, so these translations through topos theoretic bridges that you perform can really completely change the shape of results. And, and so radically introduce a new points of view on the problem you you want to investigate. Let me give you an example of this. An example I am particularly affectionate with because it's one of the first examples of bridges I discovered during my PhD studies. You are certainly familiar with the notion of completeness for for a logical theory. And you know that in general, it's it's quite hard to to to understand to prove that a certain few is complete. Now, I, I discovered the in the context of my general investigation of the crisis construction from a topos theoretic perspective that this property of completeness could be reformulated for a wide class of theories in terms of a topos theoretic invariant. Then I investigated this invariant from the point of view of other representations for the classifying topos, and it turned out that this provided completely different tools for investigating completeness. For instance, I was able to show that if you look at this invariant, which by the way is the property of a topos to be valued from the point of view of a certain kind of site, what you get is the property of the underlying category to satisfy the joint embedding property, which means that for any two objects of the category, there is a third one where you can map the two objects. So now I ask you, do you think there is a concrete relationship between joint embedding property on a category and completeness of a theory. So they look like completely different from each other. And instead, they are just the same for a wide class of theories, just because we have the classifying topos of such theories, which can be presented in two different ways. And by rephrasing this invariance in terms of the two sites on the one end you get completeness on the other hand, you have the joint embedding. And look, joint embedding is a very concrete property, unlike completeness. It is a property that if you have a sufficiently concrete category, you can easily understand when it holds or not. And indeed, I used this as a way to prove completeness for a wide class of theories. So this was one of the first applications of bridges I discovered. And certainly it was important for me because it made me realize how powerful could be this technique because you see the invariant under consideration is actually one invariant which can be automatically computed. So it's not an invariant that is difficult to compute. And even with an invariant like this, you can get such a non-trivial result. Suppose you take a more complicated invariant. So for me, it was, as I said, very, very motivating. And then of course, if you open my habilitation thesis, you will see, of course, also my papers because most of these results and most of these bridges have been published in my book, in various papers. But if you really want an overview, collecting all of them, the best places, the habilitation thesis, you will see many of such bridges. So the structure is always the same. But you will see them in different fields of mathematics. And in many cases, the insights they lead to are very surprising, very deep and hardly attainable by using more direct tools. Okay, so now we come to the main message of this talk, which is the link with morphogenesis. So in fact, when you think about what is going on in the bridge technique, you realize that actually the bridge technique exploits an essential ambiguity inherent to the concept of topos. Namely, the fact that the topos is associated in general with an infinite number of different presentations. So this ambiguity is not negatively viewed. On the other hand, it is really exploited to generate results, to generate insights. And indeed, what is going on here when you compute, when you unravel a certain invariant in terms of all this different presentation is a morphogenesis. Because in fact, what you see is that a given invariant manifested itself in completely different forms in the context of different presentations. So we have already given this example of completeness on the one hand and joint embedding on the other end. And if you change the site, you will still get other formulations, still different, different forms, etc. So we get a sort of structural morphogenesis just arising by the study of how invariants express in terms of different presentations. So if you take this technique seriously enough, this leads to a sort of peculiar way of exploring the mathematical landscape because it makes you understand that it is profitable to put at the center of the stage the Morita equivalences and the topospheric invariants, that it is profitable for you to be guided by them in carrying out your investigations because really this is the way the morphogenesis occurs. And so if you put them at the center, they will say decode for you a great amount of insights. So what is going on is a sort of rain of results that come up when you apply this methodology. So you basically start with a certain theme, you encode it in a topospheric way, and then you start creating these bridges and these bridges generate more and more insights. So the more invariants you consider, the more insights you get. And so you get all these drops falling down. And at one point they will cover the whole ground, which means that you will have a very, very satisfactory understanding of the territory in which you are. But it is a very kind of top down kind of mathematical exploration, which is a bit different from what one is used to in the specialized mathematical fields. In fact, we can depict this methodology for generating concrete results through bridges in this way. So here there is this scheme in which there is the real, say, word of concrete mathematics and the imaginary word of topos in which one makes a leap in order to be able then to descend to other concrete results which cannot be in general related directly with your starting point. So the reason why I have chosen this terminology imaginary is technical. In fact, as we shall say, topos are actually imaginary entities in the sense that they are built from mathematical theories or sites by means of completion processes which really consist in adding an infinite number of imaginary objects. In the sense of model theory. So what you do is you start with something very concrete. You try to capture its essence following Grotendick by using suitable topos. And once you get to the topos level, you switch to different presentations and by choosing invariants and computing them in terms of these different presentations you get out of the bridge. And so in the end, topos disappear and you end up just with a concrete connection between properties possibly in different fields. But you see that this leap into the imaginary was essential to perform a translation that often would have been impossible. I mean, nothing is really impossible, but very often it would be very, very difficult even to imagine such a connection. And also when you imagine it to carry it out, it can be quite complicated if you don't have this perspective. So this is more or less the structure of an application of the bridge technique. In fact, as I said, toposes are built from their presentations by means of completion processes. We have the addition of what we can call imaginaries, which I mean look to us less concrete than the entities we've started with. But on the other hand, when you build the world topos, they become objects of this topos exactly as the original ones. But you add these objects in order to arrive at a very complete environment full of symmetries. Because the more complete is the mathematical environment, the more you have symmetries and the easier are the computations. So basically, if you work, if you exploit the duality between toposes and their presentations, you have a fantastic opportunity of profiting from the simplicity of toposes presentations, the simplicity of theories or sites to make modifications to change parameters, etc. So you can profit from that on the one hand. And on the other hand, you can profit from the fact that since the toposes are very rich in terms of internal structure, they are complete in a very strong sense. Computations are generally much easier in that context, but there is a back and forth between the two levels. And so you can profit from some aspects on one hand, from other aspects on the other, and this interplay generates a lot of deep insights. Now, for those of you who are familiar with the Erlangian program, here is a slide which clarifies the relationship between this program and the methodology of bridges. In fact, the method of bridges can be seen really as a generalization of the Erlangian program because in the Erlangian program what you do is to study geometries by means of their automorphism groups. In topos theory, you actually replace groups by toposes. Actually, toposes generalize groups, so they are really something more general. And in a similar way, you study a wide class of theories by means of the symmetries which exist on these associated toposes. Okay, now I would like to talk more generally about the abstract idea of a bridge as inspired by these studies. So we have already prepared the ground for this, so I can be a bit quicker in my presentation. So basically, if we think about the problem of comparing different objects with each other, we are faced with the problem that in general the relations which exist between two objects are something very abstract. You see, so a relation is not something concrete. It is something that lives in a sense in an ideal context which is not necessarily the context, the concrete context to which the two objects that we want to relate belong to. So as we already said, it is often useful to allow oneself to introduce a further viewpoint on two objects that one wants to relate with each other and to look for bridge objects which therefore embody a certain amount of relations existing between these objects. To try to find the bridge objects that are as concrete as possible because the more they are concrete, the better it is in terms of computations arising from bridges. So we can give the general notion of a bridge object like this. So by object I mean whatever entity, so I'm using the word object in a very abstract sense. So it can be also a concept and not necessarily like a concrete object of real life. In particular, you can take some theories or opposites or whatever. So if we have two objects A and B that we want to try to relate with each other, we can think of a bridge object connecting them as an object U which admits two different representations which we call F of A, one in terms of the object A, and another G of B in terms of the object B, which allow us to understand properties of the bridge objects directly in terms of the two objects A and B. And as in the topo's theoretic setting, we can try to transfer information across A and B by means of enabling properties of the bridge object which should be invariant of course with respect to the relation which identifies these two different presentations of it. So as you can see this technique makes sense well beyond the mathematics. For instance, if you think about what you do when you perform a linguistic translation, you realize that A and B can be two different texts in different languages and F of A and G of B, they could be their meaning. And when you try to do a translation, what do you do? You put the meaning at the center and you look at how the meaning expresses on the one side and on the other. And so this is the way you actually should perform a translation. So a good translation is a translation which is invariant oriented. When you start a translation, you should fix the concepts that you want to remain invariant under the translation. So this could be of course the meaning or a certain kind of metric or musicality or whatever. You fix that and you let that guide your translation. So you see how you should put invariants at the center of the stage if you want to translate well. The other traditional approach to translation is to try to use a dictionary, but you see that this is a very partial method which can only work when your languages are sufficiently, say, isomorphic from a syntactic viewpoint to each other. If you want to translate from one language to another that is very far away, you really need to think in terms of invariants because there will not be one-to-one dictionaries. There could be a very complex expression on one hand to correspond to a simple word on the other, etc. So you cannot expect to have a one-to-one correspondence. So this is another illustration of the fact that very often you cannot relate to objects directly. You have to go to an upper level. So the bridge technique tells us something about the nature of objects as we can conceive them. In fact, an object can be thought as the collection of all its presentations. And of course, presentations are related by a fundamental equivalence relation, which is the relation of presenting the same object. And so actually we can regard any object as a bridge across its different presentations. It's important to remark that the way we access objects, also objects in real life, is by means of their presentations. We don't access the object in itself. The object in itself is actually an abstraction. An object is an equivalence class of presentations. So we access these presentations and by virtue of the coherence relations existing between these different presentations, we suppose the existence of an object. This is the sort of realist point of view. So if you believe in the existence of a reality, this is what you think. So in fact, this is the basic scientific credo that in the coherence relations. So in fact, the science has developed from the very beginning by adopting a sort of minimalist perspective in looking for explanations for phenomena, which means that when you see a lot of coherence relations between different perceptions between different presentations, you suppose that there should be something behind, that there should be something which would generate such perceptions and which therefore you could consider ontologically responsible for all these coherence relations. So this is the basic principle of scientific exploration. And in fact, the language of category theory and topos theory is particularly suitable for expressing coherence relations. In particular, the notion of shift, which is strictly related with that of topos, because as we said, a topos is a category of shifts on the site is particularly up to for expressing certain kinds of coherence relations of local global nature. And here is an example of a result, a very elementary but key result in category theory, which expresses some philosophical ideas such as that of direction of observation or generation from a source. So, in fact, there is two different senses in which generation from a source can be taken. So we have seen that an object can be thought as generating all its different presentation. So this is one sense in which an object can generate things. There is another sense, which is a bit more concrete and which is provided by the unit paradigm, which is based on the notion of generalized element. So, in fact, the unit paradigm says that we can think of an object of a category as the collection of all its generalized elements. A generalized element is simply a narrow going to that object. So it formalizes the idea of a direction of observation. So the unit embedding actually identifies an object with the function of its generalized elements. And when you see the unit embedding in the context of topos, it gives a wonderful result. It tells you that if you decide to regard a topos as a site and so to take shifts on this site, what you get is that the category of shifts, I mean the topos itself, and the category of shifts on it with respect to the canonical topology on the topos are equivalent and this equivalence is precisely given by the unit embedding. So this is a further illustration of this completeness properties of toposis which account for their centrality and for the fact that they naturally support a great deal of invariance. So again, these ideas about unification and morphogenesis makes sense in general. So whenever you have a bridge, it generates a sort of morphogenesis because a bridge indeed is precisely the expression of the connection which exists between the different manifestations of a given invariant. So examples of invariance and of bridges, of course, abounding any mathematical in any field of knowledge, not just in mathematics, you can think for instance of the notion of energy in physics. It is an invariant. So energy, if you think of it very abstractly, is an invariant. But this invariant manifests itself in very different concrete forms, thermic energy and electromagnetic energy, mechanical energy, etc. So essentially, we don't access energy in its abstract sense, but we can access energy in its concrete manifestations. And in fact, it is very natural also to use energy as a bridge for going from one for trying to transform one form of energy into another. This is just an abstract architecture in which you can interpret what is going on when you transform one type of energy into another. It is really a bridge phenomenon. So all of this actually raises some ontological questions about the nature of objects, not just in mathematics, but more generally. I mean, after all, if you think about reality, reality is, if you believe in the existence of a reality, reality is the main invariant, because we think of the existence of something which should be independent from us to a certain extent. So all of us can access certain aspects of this, this unity. And so, do you consider reality real or, well, by definition, you would say yes, but certainly it is also an abstraction if you think of it as an equivalence class of perception. So my suggestion is really that we should not be afraid of this kind of abstraction so that we obtain by adding imaginary elements, because in fact, you really get to the art of reality when you do that. And indeed, when you compute with toposis, you feel really a concreteness that maybe you would have not expected at the beginning because sometimes it's scary when you see these imaginary objects that there are a lot of psychological resistance. Think, for instance, about the root of minus one, how much resistance there was against it. But then, once it was introduced, it led to the discovery of much more symmetric environments such as the complex plane. You see, the fundamental theorem of algebra is a wonderful theorem which holds for the complex plane, not, for instance, in the real line. Whenever you complete and you add more things, you get greater opportunities for calculations, you get a higher number of symmetries, and so it is certainly something that don't necessarily bring you astray. My opinion is that really it brings you to the core, to the essence of the things. So, another couple of remarks about the relationship between contingent and universal. This actually is well formalized in the context of the bridge technique, because we have two levels, the level of the unity where invariants are defined and the level of the diversity, which is where contingency manifests itself. And so it's important not to collapse the two levels, to keep them separate and to study the duality existing between them. Every language, in fact, or every point of view, is partial. It's very important to realize about it. It doesn't make sense to look for a universal language that would be better in an absolute sense than all the others. Actually, any language has its own specific features and can be particularly profitable for enlightening certain aspects, while others could be better for other purposes. So it's like when you have different natural languages, there is no language that is better than all the others. Each language has some holes, can express certain concepts better than another, and it's only by considering the collection of all points of view of all languages that you get to the essence of things. And so, here is a final slide about this theme of completions and invariants. Indeed, it's important to take the point of view that whenever you add imaginaries and you complete something, say, concrete or partial with respect to these imaginaries, what is going on is actually to realize explicitly something which is implicitly hidden in this situation. And by performing this completion processes, you often find out that you get many more symmetries and so you can understand the phenomena much better. You see, you can have this picture. Suppose you have a highly symmetric sculpture and you break this sculpture. So suppose you have, for instance, a beautiful classical vase full of symmetries in different directions, you break it. Most of the symmetries are lost and you are left out with just fragments which call for a unity. So when you feel that you are in a situation where you are missing something that you cannot perform a certain operation, it's very good to try to complete your environment to try to reach a unity, because if you can, of course, this unity will happen at a more abstract level, but it will also make life easier for you from many points of view. Okay, so now I don't have the time to go into the future directions. So I'll skip this. But let me just mention that, well, I have a world program, of course, within mathematics for developing this theory of toposis as bridges, here is a number of things. But in fact, this technique is getting studied also by more and more researchers in different fields of knowledge, because, in fact, there are a number of topics, which I have listed here in different fields that could be profitably approached by using these techniques. In particular, in physics, there is all the subject of dualities, the different formalisms, how to reconcile different theories that apparently are incompatible, the problem of scales at which phenomena can be understood, et cetera. So topos theory has a lot to say on all of this, as well as on the other subjects that I mentioned here. And finally, a list of references for those of you who want to learn more about this. So this is a list of texts in increasing order of sophistication. So the first text is a quite a philosophical presentation of the ideas I have sketched today. Then there is another text written in French, which also makes the link between the technique of toposis and bridges and growth index heritage. Then there is the habilitation thesis, which I mentioned several times during the presentation. And finally, there is my book, which contains a lot of bridges, but it's much harder to read them as a first reference. So I suggest to follow this order if you want to learn more. So I'll stop here. And I thank you very much for your attention.