1/4 Introduction to sheaves, stacks and relative toposes
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Toposes online, 20217 / 31
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Transcript: English(auto-generated)
00:16
So it's really a great pleasure for me to introduce Olivia Caramello, who
00:22
is in University of Insubria in Como, and all the Gelfand Chair in ESOS. And I mean, will contribute so much to the theory of toposis. And she will talk about introducing, if you want, the geometric theory of toposis,
00:40
and the new theory of relative toposis, which, with respect to the logical aspect of topos theory, will surely play a fundamental role in extending the theory to higher order logics. So, OK, Olivia, it's your turn. Thank you so much for this excellent introduction.
01:01
So yes, as Alain said, this course is going to be a geometric introduction to topos theory by using the language of shifts and stacks in relation with relative topos theory. And by this, I mean doing a topos theory over an arbitrary-based topos. So authorizing a change of base topos in a similar way
01:25
as Grothendieck used to do in algebraic geometry with the schemes. Relativity techniques for schemes play a central role in the foundation of algebraic geometry. And we are trying to develop a similar formalism for topos.
01:42
So the plan of the talk is this. So I shall start this course by reviewing the classical theory of shifts on a topological space. In fact, it is quite important to take this as a starting point, also with the purpose of developing relative topos
02:03
theory, because we wanted to keep the geometric foundations at the center. And so we, in fact, one of the central ingredients of our approach to relative topos theory will be an adjunction, which provides a wide generalization of the very classical pre-shift
02:23
bundle adjunction for topological space. So it is important also to understand these new developments to start from the very classical theory of shifts on a topological space, and pre-shift more generally. Then I shall proceed to recalling the basic theory
02:43
of the shifts on a site. So we shall get to the definition of a Grothendieck topos as any category equivalent to the category of shifts on a small site. Then I will make a methodological interlude on the technique of topos versus bridges,
03:04
because it will be applied both in this course to derive results, and also in other lectures or talks at this conference, especially to derive concrete results in different mathematical contexts
03:21
by exploiting the possibility of presenting a given topos in multiple ways. So this is a sort of a basic technique that can be used for extracting a concrete knowledge from topos or more precisely from equivalences between topos presented in different ways, or morphisms
03:43
between topos, again, presented in different ways. So as when we do relative topos theory, actually we are concerned with the study of morphisms of topos, because what is a relative topos?
04:00
Well, it's just a topos which we decided to consider over another topos via a morphism connected. So basically, doing relative topos theory essentially amounts to studying morphisms between topos. And so we shall investigate the morphisms between topos
04:20
from the point of view of site presentations of topos. So we shall describe how one can induce morphisms between topos starting from functors satisfying the suitable properties. We shall see that there are two main classes of functors which induce them in a contravariant
04:41
or in a covariant way morphisms between the associated topos. These are the so-called morphisms and comorphisms of sites. In fact, the morphisms of sites represent an algebraic point of view on morphisms of topos, while comorphisms represent a geometric viewpoint.
05:00
And in fact, we shall see that they are, in a sense, dual to each other. Then I shall also review a classical notion already introduced by Grothendieck of the functor between sites, the notion of a continuous functor between sites, which in fact plays an important role
05:23
in the context of the vibrations, which are very important for developing the relative topos theory. We shall see that whenever one has a morphism of the vibrations, this gives rise to a comorphism of sites
05:41
in a canonical way, which is moreover continuous. So continuous comorphism of sites as we shall see induce a special kind of morphism between topos, the so-called essential morphisms, which satisfy pleasant features.
06:02
And so in preparation for the last part of the course, we shall describe these continuous functors also in relation with the vibrations. And then, as I said, the last part of the course, we'll give an introduction to our work in progress
06:22
with Ricardo Zansa on developing relative topos theory by using the language of stacks. So we shall take as a starting point, Giraud's paper, classifying topos, where the notion of classifying topos of a stack was originally introduced.
06:41
We shall extend many of its results and then we shall also turn our attention to relative shift topos. So we shall introduce a notion of relative site and we shall also compare our approach with the usual more classical one
07:01
that has been pursued by category theories, which is based on the notion of internal category and the internal site. So we shall see that this formalism of stacks and relative sites is much more flexible than internal categories and internal sites. It allows a parametric reasoning and it will also pave the way, as Alana has mentioned,
07:21
to an extension of the geometric logic, which is, as Lorraine explained in this course, the logic underlying the growth in the topos is overset to some higher order geometric logic provided by the fact that we can change the base topos.
07:42
And so this will allow us to quantify over parameters essentially coming from the base topos in a certain way, which is still to be made precise, but these are developments that we are pursuing at the moment.
08:01
So I will present you some of the results we have already obtained in this connection on relative topos. For the moment, we have mostly focused on the geometric side of the subject. But then once all the geometric aspects will be completely clarified, we shall also introduce a higher order relative parametric geometric logic
08:22
corresponding to relative topos. Okay, so this is the plan. And so now we can start with recalling the theory of three shifts and shifts on a topological space. Okay, so given a topological space,
08:41
what is a pre-shift on the space? Well, a pre-shift is simply a way of the functorially assigning to any open set of the space a set in a contra-variant way. So in a functorial contra-variant way.
09:02
So we want that to any inclusion of open sets corresponds a function going in the opposite direction, which in fact traditionally is called a restriction map because the idea one has when the basic example
09:23
of a pre-shift one has in mind is that of all the continuous functions on some open set of a given space. So you see that if you pass from an open set to a smaller open set,
09:40
you can restrict continuous functions on that open set to the smaller open set. And so you see that in this way to an inclusion of open sets corresponds a map going in the other direction from continuous maps on the big open sets to continuous maps on the smaller one
10:00
given just by restriction. Of course, in general, you can have a pre-shifts or even shifts which should not look at all like this. But this example of continuous functions or other kinds of functions on a space has been one of the motivating examples in the development of the theory.
10:22
And so these maps are still called frequently in the literature restriction maps. Okay, so we have said what the pre-shift is. Then of course there is a natural notion of morphism between pre-shifts which is simply a collection of maps
10:42
between the sets corresponding to the open sets which is compatible with respect to the restriction map. Now, the way I have presented pre-shifts so far is very concrete and explicit, but in fact, categorically speaking, one can simply say that the pre-shift is just a factor
11:02
defined on the opposite of the category of open sets of the space with values in them. So O of X is a post-set category whose objects are the open sets of the space and whose arrows are just the inclusions between them. So using this language, we can rephrase
11:22
what the morphism of pre-shifts is as just a natural transformation between the corresponding functions. So we have a category of pre-shifts on X which is denoted as written in the slides.
11:41
So it is basically a category of set-valued functions. Okay, so now we are going to define shifts as a particular kinds of pre-shifts. So a shift is a pre-shift which satisfies some gluing conditions. So what do I mean by gluing conditions?
12:02
Well, the idea is that one should be able to define in a unique canonical way, a certain global data starting from a set of local data that are compatible with each other. So this idea of gluing from compatible local data
12:23
is expressed formally by these two conditions in the definition of the system. So formally the condition, well, refers to coverings of a given open set of the space by a family of open subset.
12:41
So the gluing condition is formulated with respect to such covering families. So for each such covering family, one requires that whenever one has a set of elements of the pre-shift indexed by open sets in the family
13:03
which is compatible, which satisfies these compatibility conditions. So you see, this is expressed by the fact that the restrictions on the intersections of the two open sets are equal. Then there should be a unique amalgamation
13:24
of these local data and a unique global data which restricts to each of these local data. And the uniqueness of such a global data is ensured by condition one.
13:40
So condition one ensures the uniqueness, condition two ensures the existence. So together we have uniqueness and existence of a global amalgamation of a set of locally compatible data. Okay, so now as we did the four pre-shifts,
14:01
we might wonder if it is possible to categorify the notion of a shift on a topological space. Now there are a few remarks that we need to make in order to arrive at such a categorical generalization. So first of all, we remarked that the shift condition is expressed, as we said, with respect
14:23
to covering families of open sets of the space by families of open sets. So if we want to replace the category of open sets of our space with an arbitrary category, and if we want to be able to formulate a shift condition, we need to have a collection of families of arrows
14:47
going to a given object which should provide a replacement for coverings of an open set by a family of sub-open sets. This, as we shall see, will be provided
15:01
by the notion of a Grothendieck topology on a category. So this is the first ingredient. Then there is another element which requires some thought in order to arrive at the categorical generalization. It is what you see in condition
15:21
to concerning the compatibility relation. You see for the local data, which involves in the topological setting considering intersections of open sets. So of course, in general, in an arbitrary category, you will not be able to have an analog of that. But by using the categorical notion of a sieve,
15:43
we can get around this problem, and we will be able to define shifts on any category equipped with a Grothendieck topology by using this device. I shall give the details in a moment. Okay, but for the moment, I just wanted to remark that basically three shifts
16:05
can be just defined as contravariant factors on a category with the losing sets. So we don't need any additional data on the category. On the other hand, to define shifts, we need to specify a collection of the covering families,
16:21
and this will be provided by a notion of Grothendieck topology. Okay, so just a few remarks before going to the categorical generalization. So, categorically speaking, we can reformulate the shift condition
16:41
as an equalizer condition, because we can consider, so our pre-shifts have the values in sets. So in sets, we can consider arbitrary products. And so we have canonical maps as written on the slides. And so you realize immediately
17:01
that you can formulate the shift condition for a pre-shift as the condition that the canonical map going from F of U to the product of the F of UI, where the UI form a covering of U. This canonical map should be the equalizer of the two canonical maps between these two products.
17:26
And yeah, so it's important to remark that, in fact, the shift condition is actually a limit kind of condition in the category of sets. And secondly, in preparation
17:42
for the categorical generalization, it's important to remark that by using the technical device of SIFs, we can avoid referring to intersections for formulating the gluing condition. So how does one do this?
18:01
So given a covering family F of open subsets UI of an open set U, you generate a SIF starting from that, which in this case will be the collection of all the open subsets of U, which are contained in some UI.
18:22
And so by doing this, you can basically rephrase the compatibility condition by requiring instead of a family of elements indexed by the open sets in the covering family,
18:41
you take a family of elements indexed by all the open sets in the SIF. And the compatibility relation becomes the relation that whenever you have an open set W prime included into W,
19:03
you should get the equality between the value at W restricted at W prime and the value at W prime. So you see that in this way, you have eliminated the reference to intersections.
19:21
And so you can see that by using this idea of SIFs, of taking everything which is below, everything which is essentially generated by composition on the right from a certain family of arrows, because you can regard the disinclusions as arrows in the category of open sets of X,
19:41
you can avoid the referring to intersections. And so you can already understand why it is possible to define shifts on an arbitrary category equipped with an essentially arbitrary notion of covering families in it, as it will be provided by so-called Grothendieck topology.
20:03
Okay, so now we shall go in more details about all of this. But before this, I would like to give some examples of the shifts. So I have already mentioned the main motivating example provided by continuous functions on topological space.
20:21
But of course, shifts are used in many other areas of mathematics in the context of the differential geometry analysis, differential algebra geometry, et cetera. So for instance, you have shifts of regular functions on the variety of differentiable functions
20:41
and the differentiable manifolds, allomorphic functions on a complex manifold, et cetera. So in fact, in mathematics, shifts arise in many different contexts. Very frequently, shifts appear as endowed with more structure than just the set theoretic one.
21:02
In fact, for instance, in algebraic geometry, one has the shifts of modules or shifts of rings, even shifts of local rings, et cetera. So far, we have talked about the shifts of sets. And in fact, there is a good reason for taking as a starting point the shifts of sets
21:24
rather than shifts of more complicated type structures or even more general structures. In fact, Grothendieck himself realized about the importance of first talking about the shifts of sets in order to have a better categorical properties
21:42
when you consider the whole category of shifts of sets on a topological space or more generally on a site. Because if you replace the sets with another category, in many cases, you lose some pleasant categorical properties.
22:01
So it is good to define categories of shifts of sets and then try to understand shifts of more complicated structures as relative to these shifts of set valued structures.
22:23
So formally, the way this is done, at least for geometric theories in the sense which has been explained by Lorang in his lectures is you look for instance at a shift of the models of a certain geometric theory.
22:40
In particular, it could be shifts of modules or shifts of rings, shifts of local rings, et cetera. You regard this as a model of the theory of such a structures. So a model of the theory of models of rings, of local rings formalized within geometric logic inside the category of shifts of sets
23:03
on the space or the site. So this is the way we can naturally deal with these shifts of more complicated structure. So yes, this is an important remark
23:21
to make from a formal viewpoint. Okay, now let's proceed to talk about this fundamental adjunction between pre-shifts and bundles on a topological space which I mentioned in the introduction of my talk.
23:43
So after this, we shall describe shifts on a cycle. Okay, so how does this very classical adjunction work in the topological setting? So first we define, given a topological space X,
24:01
a bundle over S simply as a continuous map towards X. We have, of course, a category of bundles which is simply the slice category. So the top here denotes the category of topological spaces and continuous maps. And here I am taking the slice category over X.
24:21
So then the objects are arrows going to X, continuous maps towards X and the arrows are commutative triangles. Okay, now why is it interesting to consider bundles in relation with shifts?
24:43
Well, because there is a very nice construction which allows us to build a shift from an arbitrary bundle through the consideration of the so-called cross sections of the bundle.
25:01
So given an open set of our space X, a cross section over this open set of a bundle is simply a continuous map defined on that open set going to the domain of the bundle, such that when it is composed with the bundle map,
25:21
it leaves the inclusion of the open set into the space. So of course you can collect all of the cross sections of a given open set in a set. And this gives a set, the set of all cross sections
25:41
over the given open set U. And if you think a minute about this, you realize that this operation is functorial in a contra variant way in the open set, because if you switch from an open set U to an open set V contained in U,
26:03
you get the restriction operation because you can restrict across the section over U to across the section over V. This is a completely clear. So in this way, you actually get a functor, a contra variant functor on the category of open sets of X, the V values in set,
26:22
namely a pre-shift. And it is not hard to show that this pre-shift is actually a shift. And this is called the shift of the cross sections of the bundle T. So given the fact that we can build a shift from bundles, it is natural to wonder
26:43
if one can go in the other direction as well. And this is possible through the construction of the so-called bundle of germs of a pre-shift. So suppose that you started with a pre-shift on a space X,
27:02
you can build a bundle out of this by considering germs of sections of the pre-shifts at the points of the space. So first we have to define what a germ is at a given point of the space.
27:21
So a germ is an equivalence class of the sections defined over open neighbors of the point. And the equivalence relation is what I have written in the slides. So two sections are equivalent in an open neighborhood.
27:46
Of course, the open neighborhoods can be different. So they are considered equivalent if there is some open neighborhood of the point contained both in U and in V on which their restrictions agree.
28:03
So of course, this is an equivalence relation. So we can take equivalence classes and we can do this for each point of the space. And so if we fix the point and we take the collection of all germs at this point, we get what is called the stock of the pre-shift
28:22
at the given point. And of course, we can consider the disjoint union of all the stocks. So it is a union indexed by the points of X. And of course, we have a projection map to X
28:42
which is defined in the obvious way. So it takes just the point on which the germs are defined. And we can topologize the domain of this projection map in such a way that it becomes a local polynomial.
29:05
And so what we get is actually what is called the bundle of germs of the given pre-shift. Okay, so now we have two constructions, one going in one direction, the other one going in the converse direction.
29:21
And in fact, what happens is that, so we have these two factors, gamma, which is the factor of the cross sections and lambda, which is the factor giving the bundle of germs.
29:42
And in fact, these two factors form an adjoint pair. So lambda is adjoint on the right, on the left, sorry. And gamma is actually a global section, well, it's a section factor. So in particular, if you can apply it
30:01
to the space itself, in this case, you get the global sections of the bundle. And so in fact, a key result is that this adjunction actually restricts to an equivalence of categories where you have on the one hand shifts,
30:24
so the pre-shifts which satisfy the shift condition. And on the other hand, some particular bundles which can be characterized as being the etal bundles or also called the local homomorphism. So in fact, the restriction of this adjunction
30:43
is what you get by restricting to the fixed points, fixed points of the adjunction. So in general, whenever you have an adjunction, you can restrict it to an equivalence of categories by restricting to the fixed points. So this is a general process. And if you apply it in this case,
31:00
this is what you get. So this is very nice because it allows us to geometrically think about the shifts as particular kinds of bundles, namely the etal bundles. This has several pleasant consequences concerning the geometric understanding of a number of constructions on shifts and pre-shifts.
31:26
So in particular, I would like to point out two of these nice insights that such an adjunction brings out. So first of all, the shiftification process. So if you take most books on Topos Fury,
31:43
you will see that the shiftification is described by using the plus plus construction, which is a technical means of constructing this, but not necessarily very geometrically intuitive. So thanks to this adjunction,
32:01
one has a more geometric understanding of the shiftification process. Because in fact, we can think, we can describe the shiftification of a given pre-shift as simply the result of applying the successively the two factors forming the adduct.
32:22
So basically the shiftification of a pre-shift is simply given by the shift of cross sections of the bundle of germs of the pre-shift. And you see that this is geometrically much more satisfying because you really get a geometric substance.
32:42
You get the geometric understanding of what the elements of the shiftification really are. So it's not just the formal quotient construction, but you have a geometric realization of such elements. We shall come back to that because in fact, we shall be able
33:02
to get the generalization of this in the context of an arbitrary style and even in the context of stack. So just to keep in mind for the moment, these features because we shall then provide the generalization in the categorical and stack theoretic setting.
33:23
Okay, so we have talked about the shiftification geometrically understood. There are also other advantages of this point of view of shifts as a tall bundle. For instance, suppose you have a continuous map between topological spaces. And suppose you want to describe the effect
33:42
on shifts of such a continuous map. Well, of course, there is a direct image of shifts, which is defined in a strike forward way, basically just composed with the action of the continuous map on open sets of the two spaces.
34:02
So this is completely straightforward, but suppose you want to understand the inverse image of shifts along this continuous map. Well, this is not completely straightforward. If you want to do it in the language of shifts, we shall see that you can do it
34:22
by using the can extensions for instance, but here by using the identification between shifts and the type bundles, you have a very nice simple description because taking inverse images of shifts on Y along F corresponds precisely to take the pullback
34:41
along F of the etal bundles corresponding to these shifts. So in fact, one can show that taking the pullback of an etal bundle still gives an etal bundle. And so this is how the inverse image operation of shifts actually works. So you see that really this junction
35:03
brings some very nice geometric intuition in the picture. So it has been a question for several years, whether one could find a good analog of this working for arbitrary sites or even possibly extending to stacks. And in fact, in our joint work with Ricardo Zappa,
35:25
we have indeed provided such a generalization, not just for a prefix, but more generally for index category. So this will be described in the last part of the course. Okay, so now that we have talked about this fundamental junction,
35:42
we can go to the categorification of shifts. So how to define shifts on an arbitrary site. So for this, as I already anticipated,
36:01
it is necessary to talk about sieves because you see sieves were fundamental for giving a notion of compatible family of local data without requiring the intersection problem. You remember about that. So it is a technical device that is essential for defining shifts in the general categorical scene.
36:23
So formally, what is a sieve in a category? Well, given a category and an object of this category, a sieve in the category on that object is simply a collection of arrows in the category towards that object,
36:40
which is closed under composition on the right. So the condition is that whenever an arrow is in the sieve, the composite of this arrow with an arbitrary arrow should again be in the sieve. So you see, this is a categorical generalization of the condition we had for topological spaces. We wanted the sieve to contain any other smaller open set, you see,
37:04
and here we get the condition that it should be closed under composition on the right, which is just the categorification of it. Okay, so sieves are very nice objects, in fact, because you can make a lot of operations on sieves. In fact, you can understand the sieves more abstractly
37:25
as the sub-objects of the corresponding representable functor in the category of pre-sieves. So this is an abstract understanding of sieves as a sub-functor of representable functors, but you can avoid that point of view.
37:41
You can reason about the sieves in a perfectly concrete way. So in particular, you can compute pullbacks of sieves, which give rise again to sieves. So here is the operation. So if you have a sieve on an object and an arrow going to that object, you can pull the sieve back along this arrow
38:05
to get another sieve. So this sieve consists of all the arrows, which when composed with the given arrow belong to the sieve. So it's a very naturally defined operation. And in fact, it really corresponds to taking a pullback
38:21
in the corresponding pre-sieve topology. But as you see, you have a perfectly concrete description of this operation without involving pre-sieves. Okay, so now that we have talked about sieves, we can introduce the fundamental notion of a Grothendieck topology on a category,
38:43
which will be the basic setting for us to define sieves. So a Grothendieck topology is a way of assigning two objects of the category, a collection of sieves on those objects
39:02
in such a way that some natural conditions are satisfied. So the first condition is called the maximality axiom. So it requires the maximal sieve on each object to be in the topology. So of course, this is quite intuitive.
39:21
So of course, the maximal thing should be covering. It's quite natural. Then we have a second axiom, which is quite important. It's called the stability axiom. So it requires that the pullback of any covering sieve should be covering.
39:40
Again, this is quite intuitive. In fact, in the topological setting, it corresponds to the fact that if you have a covering of a certain open set by a family of open subsets, when you pass from that open set to a smaller open set, and you intersect each of the open sets in the family
40:02
with that one, you still get a covering of the smaller. So this is stability. And so you see that in the topological setting, it is satisfied. So it is natural to require it as well on an arbitrary category. Then you have the so-called transitivity axiom,
40:21
which says that whenever you have a sieve such that the pullbacks of this sieve along all the arrows of the covering sieve is covering, then the sieve itself should be covering.
40:40
Okay, so this axiom is a bit less important than the other ones, especially then, well, the maximality axiom, you can always make it hold without any problems. But the stability axiom is really the crucial axiom to have. In fact, for defining sheaves, basically you just need the maximality and stability.
41:06
And because then you can generate the growth and the topology starting from those, and it will have exactly the same sheaves. So basically not all the axioms of growth and the topology have the same status.
41:21
In fact, the most important one is really the stability axiom. In any case, I mean, one often works with the basis for growth and topologies or some smaller presentations for growth and topologies. And in general, it is important to dispose of techniques
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for computing growth and topologies presented by some families of sieves. There are formulas for this. In my book, for instance, you can find the formula for computing the growth and the topology generated by an arbitrary collection of sieves
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and some other techniques for computing topologies starting from basis satisfying certain properties. So in general, it is an important theme that of being able to compute a growth and the topology starting from certain sets of data. From a logical point of view, in fact,
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as Laurent will explain in the last part of this course, being able to generate a growth and the topology starting from a collection of sieves corresponds to deriving theorems within geometric logic starting from certain axioms. So as you can see, it's something quite significant,
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especially when you can really achieve a full description of the growth and the topology. It means you have a classification of all the geometric logic theorems that are provable in a given theory. Okay, in any case, so the sieves which belong to the topology
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are called J-covering, where J is the topology. And then a site is defined simply as a pair consisting of a category and a growth and the topology. So which kind of sites in terms of size
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are we going to consider? Well, for defining growth and toposites, one restricts to small sites. And by this, I mean that the underlying category of the site should be smaller to have just a set of objects and arrows. But for technical reasons,
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it is important also to consider larger sites, which still can be studied, can be associated with smaller sites in a meaningful way. So these are called small generated sites.
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And in fact, the site is said to be small generated if the underlying category is locally small and admits a small J dense subcategory in the sense expressed in the slides. So in particular, it will be convenient
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for a technical viewpoint to consider a topos itself as a site with a topology called the canonical topology. And you see that a topos in general will not be smaller but such a canonical sites will always be small generated
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by definition of a topos as a category having a smaller set of generators. So it's important to keep in mind that while a growth index topos is formally defined as a category of sheets on a small site, in fact, one can extend to all small generated sites
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without changing the resulting categories of sheets. Okay, but this is just a technical point on which we shall come back later. Okay, now let's give a number of basic examples of growth index topologies to get you familiar with the concept.
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Of course, we can always put the growth index topology on any category by taking as covering seeds just the maximum ones. Then there is a very nice interesting topology that one can put on an arbitrary category. It's called the dense topology and is defined by taking as covering seeds
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precisely the stubbly non-empty ones. So the ones such that they are pulled back along arbitrary arrows is always non-empty. So this simplifies in the situation where the category satisfies the right or condition, namely the property that you see displayed in the slide.
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The fact that for any pair of arrows will become co-domain, you can complete them to a commutative square. So under such a hypothesis, in fact, the pullback of any non-empty seed is again non-empty. And therefore, the dense topology
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specializes to the so-called atomic topology, whose covering seeds are precisely the non-empty ones. The atomic topology is very important for several purposes in toposphere, in particular in connection with the toposporic interpretation of Galois theory and its categorical generalizations.
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Okay, other examples of topologies. Well, of course, the motivating example for us was sheaves on a topological space. So in fact, we were considering pre-sheaves and sheaves on the category of open sets of the topological space. On such a category,
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there is a canonical Grothendieck topology one can consider. So in fact, we postulated that the families that should be covering be exactly those which give covering families in the usual topological sense. That is, the open set should be the union
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of all the open sets in the family. This, of course, can be generalized to pointless topological spaces, also called the frames or complete eigen-algebras. So the frame is a complete lattice
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in which the infinity distributive law of arbitrary joints with respect to finite needs holds. So you can really see this as it were, the lattice of open sets of a topological space, even though in general, you might have frames which are not of that form. Only the spatial frames come from topological spaces.
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There are many other frames which are not of this form and which can be studied and which can be interesting in their own right. And of course, on such a frame, one can define a Grothendieck topology by using joints in this frame, thanks to the fact that these arbitrary joints distribute over finite needs,
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which ensures that the stability axiom for Grothendieck topologies is satisfied. So the topology we define in this way on a frame is called the canonical topology on the frame. Okay, now another set of examples of a different nature.
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So given a small category of topological spaces, which is closed under finite limits, typically one supposes that, and undertaking open sub-spaces, there is a natural topology one can define on it called the open cover topology
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because its covering families are precisely given by families of open embeddings which cover the given space. So in the sense written there. In fact, the open cover topology plays an important role in the construction of Grohe and Petit toposies in the topological setting.
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In fact, with Ricardo Zampa, we have introduced a higher analog of this open cover topology on the category of toposies itself. And in fact, we have shown that thanks to this, one can essentially regard any Grothendieck topos
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as a sort of Petit topos associated with a very big topos related to that by a local morphism by attraction and et cetera. And so in fact, this idea of the open cover topology is an interesting one.
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So I will not have the time in this course to talk about this result, but if you are interested, you will be able to read about it in our forthcoming work. Okay, another very important example of Grothendieck topology is the Zariski topology,
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which can be defined on the opposite of the category of finitely presented or equivalently finitely generated commutative unit. So of course, this topology plays a key role in algebraic geometry, and it admits a very simple intuitive definition.
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So the covering the cursives for this topology. So I talk about cursives because I switched from the opposite of this category to the category itself. These are those which contain,
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which contain finite families of localizations of the given ring at elements of the ring. Families which are characterized by the property that the ideal generated by these elements is the wall ring.
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So equivalently, this collection of this set of elements is not contained in any proper ideal of the ring. Of course, you understand the geometric significance of this definition. You see, if you think of the Zariski spectrum
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of ring, you can see that this in fact corresponds also to a more intuitive kind of covering relation at the topological level in terms of the spectrum.
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Okay, then finally, Laurent has talked about syntactic sites in his course. And of course, these are very important kind of sites that one can build from any kind of first order geometric theory.
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In fact, depending on which fragment of geometric logic you consider, you have different versions of the syntactic sites. So if the theory is regular, for instance, we have a regular syntactic site. If the theory is coherent, you have a coherent syntactic site. If the theory is geometric, you have geometric syntactic sites. So different versions of syntactic sites, but which will present always
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the same classifying topics. And so you have different ways of say, embodying the syntax and the proof of theory of a theory in a site, which actually presents its classifying.
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In fact, it's quite interesting also to compare properties related to different fragments in which a given theory can be considered. There are compatibility relations existing between different fragments.
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And for instance, you can understand them very well by using the bridges, because you have just one classifying topos and different presentations for it, provided by these different fragments. And the point is that you can understand several invariants from these different points of view,
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and they will give rise to such a compatibility relation. So in any case, this is just a remark. So if you want to know more about this, you can take my book and you will find several results, several compatibility results of this kind approved through bridges.
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Okay, so now we are ready to introduce SHIBs on a site. So SHIBs are defined in the obvious way. So simply as a contra-variant functors with the values in sets defined on the given category. Then for defining SHIBs on a site,
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we have to talk about compatible families of local data indexed by covering families in the Groton Dictropology. So we define a notion of matching family for a SIB of elements of appreciative. So this is defined as a way of assigning
54:42
to each arrow in the SIB an element of the receiver in such a way that this compatibility condition, you see, is satisfied. You see that here the SIB condition is fundamental because you see that here I am considering the composite of F with G.
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So I am using the fact that since I have a SIB and F belongs to the SIB, also F composed with G belongs to the SIB. And so it makes sense to consider that element because I should have an element for each arrow in the SIB. And so I can formulate this compatibility condition. And so this is what a matching family is.
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And then for such a family, we can define what should be an amalgamation. So an amalgamation should be a single element of the presheaf at the given object, which is sent by the presheaf to all this local data
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along the arrows of the SIB. So you see everything is very natural, very unsurprising. Okay. So again, as in the topological setting, we can formulate the shift condition in terms of the equalizer.
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And so by considering all the presheafs on a category, which are shifts with respect to a given Grothendieck topology, we get the category, the category of shifts, which is what is denoted like this.
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So shifts on Cj will denote the category of shifts on the given site, and as arrows, the natural transformation between these shifts regarded as presheafs. Just to remark, the shift condition can be expressed categorically in a very nice way.
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So you see, I mentioned that a SIB can always be considered as a sub-object of the corresponding representable. So you see this vertically in the triangle. And the shift condition can be formulated as a sort of extension condition.
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So a matching family for a given sieve can be thought as a natural transformation from the given sieve to the presheaf. This is quite clear because you see the compatibility condition is amounts precisely to naturality.
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And so you see that the shift condition by the unit dilemma can be formulated by saying that every natural transformation defined on a covering sieve admits a unique extension as in this diagram.
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Okay, so finally we can define what a Grothendieck topos is. So a Grothendieck topos is any category which is equivalent to the category of shifts on a small site or more generally a small generated site. Okay, so we have got to the central notion of this course.
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Examples of topos is, well, here are just three classes of examples. Of course, there are infinitely many examples, but I selected these three just to show you very quickly
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how general topos are. Because you see, the first example deals with categories. So whenever you have a category, you have an associated topos, the topos of presheafs on that category, which you obtain by keeping the category with the trivial topology.
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On the other hand, if instead of a category you start with a topological space, you have also a topos associated with that, the topos of shifts on the topological space. Also, you can decide to start from a group and to consider the category of actions
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of the group on discrete sets. So you can also take the group to be topological if you want. And in which case you take the continuous actions of the group on discrete sets. And you can show that in this way,
59:20
you indeed get Grothendieck topos because you can present this topos as the topos of shifts on a particular site. So the site was underlying the categories that of non-empty transitive actions and the topologies, the atomic topology, which we introduced here.
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So you see already these three basic examples to show you that topos extend categories, topological spaces and groups. So given the fact that all of these concepts play a central role in...
01:00:00
mathematics nowadays you can understand why toposies have a very big potential to have an impact in essentially across all mathematics because they simultaneously generalize all of this and there is much more to that because of course as we shall see toposies can be attached also to other kinds of entities. Lauren has talked about how to associate
01:00:24
toposies with theories etc and there are still many other approaches to the construction of toposies that one can introduce.