 So, yesterday we gave the fundamental definition of SHIF on etopological space and with the purpose of getting to the categorical definition of SHIFs on a site, we introduced the language of SHIFs. So, let's recall that a SHIF on an object in a category is a collection of arrows to that object which is closed under composition on the right. As we remarked yesterday, this idea of closing a given family of arrows to a given object by a composition on the right has the advantage that it makes possible to express the condition for a given family to be compatible without requiring the intersections to exist as in the topological case. So, yesterday we made this remark which actually is what we need to define a notion of matching family in the categorical setting and then a notion of SHIF in the same setting. Okay, so let's introduce the fundamental definition of grotendictopology on a category which is what is meant to replace the notion of covering of an open setting etopological space by a family of open subsets. So, a grotendictopology on a category is just a way of assigning to each object of the category a collection of SHIFs on that object in such a way that some natural conditions are satisfied. These conditions are the following. First, we require the maximality axiom which is simply the assertion that the maximal SHIF on each object should stay in the topology. Then we have the pullback stability axiom which says that any pullback of a SHIF belonging to the topology should again belong to the topology. Again, it doesn't need a lot of justification because it is a very natural requirement. In fact, yesterday we gave the definition of pullback of a SHIF along the certain arrow. So, if you have a SHIF S on an object C, you can pull this back along H to get a SHIF on D which will consist of the arrows that when composed with H belong to S. So, the pullback stability axiom refers to this construction. So, in order to concretely understand this in the setting of topological spaces, let's just postulate that when we have X topological space, we can naturally define a Rotten Dick topology on the category of open sets of the space which we introduced yesterday by saying that So, this will be called the canonical topology on this category. So, we postulate that a given family of arrows in this category going to a given object. So, to give an open set. So, just a family of arrows will be, since arrows are inclusions, it will be just a collection of inclusions of sub-open sets into a given open set. So, then we postulate that such a family belongs to the topology if and only if the open set U is actually covered by the UI in the classical topological sense. And so, you understand the pullback stability condition holding in this particular case because it amounts precisely to the infinite distributive law because you see if you have a sub-open set V of U and U is covered by a family of open subsets UI Actually, when you pull this back, what you will get is the family of these intersections which certainly cover V because of the fact that this infinite distributive law. From this you understand that actually the pullback stability condition is a natural condition but it is not a trivial one because it requires a certain distributivity to hold at least in this particular case. In fact, there is a third axiom in the definition of Rotten Dictopology which is what you see there in the slides, the transitivity axiom which is again quite natural because it says that if you have a certain sieve R such that all its pullbacks along the arrows in a sieve which is in the topology are in the topology then the sieve itself should be in the topology so it is a sort of converse to the pullback stability and in fact this axiom is less fundamental than the pullback stability axiom because in fact as far as sieves are concerned if you start with a collection of sieves in the category which satisfies the maximality and pullback stability conditions you can always generate a Rotten Dictopology out of it taking the intersection of all the Rotten Dictopologies which contain this and when you look at the sieves as we shall define at any moment with respect to one or the other they will be the same so it means that actually the two key conditions are really well the maximality condition which of course is quite trivial but the pullback stability condition is really crucial so the sieves which belong to the given topology will be called the covering sieves for this topology so we shall say that the sieve is J covering if it belongs to J so now we can define the notion of site as simply a pair consisting of a small category and a Rotten Dictopology on it and this will be the basic categorical settings for defining sieves in full generality before giving this definition let's just present a few basic examples of Rotten Dictopologies well of course what you can remark is that for any small category you can always define a Rotten Dictopology on it in a trivial way just by taking the maximal sieves on all the objects of the category of course all the axioms are trivially satisfied this actually is important because when we shall define sieves on a site we will see that the sieves on trivial sites I mean sites where the topology is trivial they are just pre-sheves so one gets pre-sheves by taking sheves with respect to the trivial topology then there is a more interesting topology that one can put on any category it's called the dense topology and it's defined by saying that sieves which cover are precisely the stably non-empty ones so by stably non-empty I mean this condition here holds which is the same thing as saying that these are sieves such that their pullback along any arrow is non-empty now things simplify if our category satisfies the right or condition which is the property that any pair of arrows with common codomain is completed to a commutative square because in this situation one immediately sees that the pullback of a non-empty sieve is again non-empty and so the definition of the dense topology simplifies and we get what is called the atomic topology which is defined as the topology whose covering sieves are precisely the non-empty ones this will be useful for us for understanding, for instance, Galois theory from a toposteoretic perspective so we shall come back to this in tomorrow's lecture ok, now let's go on with other examples of growth and dik topologies of course we have already remarked that when we had a topological space there is a natural growth and dik topology associated with it on the category of subsets which is what I have written here so it is actually the canonical topology on this category one can define a notion of canonical topology on any category which is the largest coverage for which all the representables are sheets and so in this particular case we get that topology now more generally yesterday we have talked about pointless topology and the fact that topos theory actually generalizes pointless topology rather than classical topology in the sense that when you define sheets on topological space you basically forget about the points of the space and you only focus on the lattice theoretic structure of the category of its open sets so since you don't actually use the points what you could do if you start with just a frame which is a complete lattice where the infinite distributive law holds of arbitrary joints with respect to finite needs of course you can also define a growth in the topology on such a frame which is also called the canonical topology and which is defined like this so we say that a given family of elements below a given element a covers it and only if the joint of all these elements is equal to the given element and in fact what makes full depth stability condition satisfied is precisely the infinite distributive law which holds on a frame just by definition of a frame in fact frames are the basic objects of study in pointless topology because in fact you remark immediately that the category of open sets of topological space is a frame so in fact frames can be regarded also from a logical viewpoint as a complete hiding algebras because in fact you can immediately see that on any frame you have for any pair of elements a and b an element which is denoted like this which satisfies this universal property that an element is less than or equal to this even only if you have that so this is of course an adjunction property and so you see here I have used this notation of the implication because in fact it makes sense to interpret the equitizmistic propositional logic disconnected in the equitizmistic propositional logic as this operation in a frame so things become sound when you do that but we shall come back to this point later when we talk about the internal logic of a topos which in fact is precisely due to a hiding algebras structure which exists on all sub-object lattices in the topos ok so for the moment let's continue with other examples of grotendic topologies suppose you have a small category of topological spaces which is closed under finite limits and undertaking open subspaces then in such a situation it is next round to define a grotendic topology called the open covered topology by specifying as a generating basis the collection of open embeddings which cover a given space ok so then of course I couldn't miss a fundamental example of grotendic topology which plays a crucial role in algebraic geometry the zariski topology which for instance can be defined this is the basic example of course you can define it over an arbitrary ring but here let's just give the definition for the basic ring z so we consider the opposite category of finitely generated commutative rings with unit which is the same thing as finitely presented commutative rings with unit and we postulate that a sieve in this opposite category or here I have written co-sive in the category of such rings if and only if it contains a finite family of canonical maps of this particular form so this notation a f i indicates the localization of the ring a at an element f i so I take all finite families of arrows of this form and take elements f i, f n which are not contained in any proper ideal of the ring so it means that they generate the ideal generated by all is the world ring so you can easily see this is indeed a grotendiktopology and so the topos which you get by taking sheaves on this side is called the zariski topos of course there are many other grotendiktopologies that you can put on the same category and which also are relevant for many other purposes in geometry, in fact in general if you fix a category there exist in general infinitely many grotendiktopologies that you can put on it so you have a great degree of freedom because you see in the notion of sight you have a basic category so it means that you have two degrees of freedom given by objects and arrows and then when you put a grotendiktopology you add a further degree of freedom which makes you understand how general the notion of sight is and so it makes you understand that you can find sights basically in any area of mathematics because of course categories can be found everywhere and of course in many situations you can find suitable grotendiktopologies to put on them and of course all of this becomes interesting when you pass to the category of ships on these sights and you look at the invariance of this category of ships you are able to capture some important property written in the language of the original sight that you are interested to investigate and so in this way you are able to exploit the fundamental duality existing between sights and purposes but we shall come back to that especially in tomorrow's lecture ok so now let's give the formal definition of ship on a sight so first as we did in the topological setting we define what a pre-ship is so a pre-ship on a small category is simply a contravariant factor from the category to the category of sights then we have to define what a compatible family of elements of a pre-ship is so we give the definition of a matching family of elements of a pre-ship index by the arrows of a covering seam by saying that we should have an element XF indexed by any arrow F in the sieve and that we should have this compatibility condition that so basically we we require so if we have F in our sieve we want to have an element and we want that if I consider the composite of my arrow with another arrow because remember that we are in the context of seas so when we compose an arrow in the sieve with an arrow on the right the composite arrow should again be in the sieve and so I should have another element corresponding to that because I should have one element for each arrow in the sieve so if I have an element corresponding to F I should also have an element corresponding to F composed with G and this of course will belong to that and you see the compatibility relation is that you see if we look at what happens at the level of the pre-ship here we have X of F here we have X of F composed with G and of course I can apply the pre-ship to that and the compatibility condition is this equality so you see this is a very elegant way to get rid of the problem of considering intersections and you actually don't need intersections to express the compatibility condition the only thing you need is to consider things that can be composed with all the arrows in your covering time this is the crucial remark which allows us to consider covering notions on arbitrary categories not just categories where you have finite limits so now we know what an etching family is then we define the notion of an amalgamation of such a family and this is as in the topological setting defined as a single element of this which is sent by the pre-ship to all the elements indexed by the arrows in the covering so you see all of this is very natural and so of course all of this allows us to define ships as the pre-ships such that every matching family indexed by any covering seed in the topology has a unique amalgamation so we require in a completely analogous way as what was happening in the topological setting the existence of the possibility of gluing things along covering families provided that the compatibility conditions are satisfied then Glutenbeak decided to consider ships in an isolated way one from the other but to consider them all together mainly to construct the category of all ships on a given site and this was really a great idea because of course the resulting category will be in general quite big but the great advantage of considering these categories of ships is the categorical properties that they satisfied we already saw this yesterday in the topological setting that all the categories of ships of sets on a topological space possessed very pleasant categorical properties they were both complete and co-complete so they had all small limits because many problems that one encounter in mathematics are actually due to the fact that the structures with which one works often had holes they don't allow certain operations to be made and so you see when you get to the topos liberal basically all computations become possible very rich categorical structure and so already you see in the topological setting it was a very interesting metamorphosis to use Glutenbeak's word from the classical notion of topological space to the notion of category of ships on a space because you see a topological space by itself is not something with a lot of structure while by passing from a topological space to its associated category of ships you get all this enormous categorical structure and so you can make a lot of computations which then you can try to reinterpret from the point of view of the topological space which served for generating the topos so you see you have two levels the level of size I mean the setting of topology and the level of toposis attached to this size on which invariants are naturally defined and then it becomes very interesting to study the duality which exists between the two levels in particular we shall see that the key ingredient of the bridge technique is precisely the study of how invariants defined the topospheoretic level expressed themselves in the context of different sites or more generally presentations of the toposis okay so a grotesnic topos is defined formally as any category equivalent to the category of ships on a site so this is a constructive definition of course as we mentioned already yesterday there is a fundamental ambiguity in this definition in the sense that one in principle could have different sites giving rise to equivalent toposes but this ambiguity shouldn't be seen as a negative element because in fact as we shall see especially tomorrow it formalizes precisely the phenomenon which occurs when different languages are used to describe the same structures so it expresses a very general form of mathematical duality or equivalence and so in fact it is really possible to exploit this ambiguity in a fruitful way so all the bridge technique in fact will be about that okay so now before going to that let's remark that in fact the notion of topos is quite powerful because it generalizes in a very natural way fundamental notions which we are extremely familiar with such as categories topological spaces or groups in fact we have already seen that for any small category topos attached to that which is just the topos of three ships on this category which we can regard of course as ships with respect to the trillion topology in fact this topos could be seen really as an extension of our category in fact it is the free co-limit completion of the category and so to a certain extent this topos is actually generalized categories in the sense that you cannot quite recover the category from the associated topos but almost in the sense that you can recover the co-ship completion of the category from the topos as the full subcategory on its irreducible objects so the co-ship completion is actually a quite innocent operation because it corresponds precisely to splitting all idempotence in the category so for instance if you have a category with finite limits it will be automatically co-ship continued and so you see that just up to co-ship completions really we can identify categories with the corresponding topos and then of course we have already remarked that to any topological space we can attach a topos which is just the usual category of ships or sets on it this as we said yesterday doesn't strictly generalize topological spaces because in general one cannot recover a topological space from the corresponding topos unless the space is sufficiently separated so yesterday we considered these categories of ships of sets of topological space and I already remarked that you cannot in general recover a space from the associated category of ships unless the space was sufficiently separated the formal definition is a sober if every irreducible closed set is the closure of a unique point so in fact most of the topological spaces which naturally arise in mathematics are already sober such as compact household spaces or any underlying space of a scheme is sober so if you work in algebraic geometry you don't have to worry about this this is due to the fact that in fact what is not possible to recover is not the open sets of the space so yesterday the open sets could be recovered as subterminal objects in the topos but what you cannot in general recover is the points of the space because yesterday we showed that any point of the space gave rise to an adjunction to a point of the adjunction with the property that the left adjoint preserved the finite limits and so we shall see later that this gives an ocean of points of the corresponding topos but actually there could be other points there could be other points in the topos which do not come from the points of the space and also if the space is not sufficiently separated in fact there could be not enough open sets to separate distinct points so this is why in fact toposes are not strictly generalizations of topological spaces but they are just a generalization of sober topological spaces or if you want to think in terms of pointless topology they are a generalization of frames because of course when you consider a frame f then you put the canonical topology on this frame basically you can recover the frame from that still by taking subterminal objects as in the topological setting and so in this way you get really an identification between a frame and the corresponding topos in fact the frames when you decide to regard them from a topological viewpoint are also called locales and so what one says is that toposes of this form which are called localic toposes actually generalize locales or the associated localic toposes exactly the same ok now toposes actually generalize discrete groups but more generally in fact you can build a topos out of any topological group by considering the continuous actions of the group on discrete sets so actually you can see that this is a topos by exhibiting a site for that which is actually an atomic site because so the category of continuous actions of the topological group on discrete sets is denoted by this when you take the full subcategory of that on the non-empty transitive actions you actually get a site of definition for that because you can show that the topos is equivalent to the category of shins on this subcategory with respect to the atomic topology which we introduced earlier of course in general the equivalence which identifies two topological groups when they have equivalent associated toposes it becomes a trivial in the discrete case but not in general but in the topological groups this relation is called moritequivalence and it is by no means trivial in fact we shall see some interesting examples of that ok, so now let's resume the basic properties of this categories of shins well exactly as in the topological setting we have that the inclusion of shins into pre-shins admits a left adjoint which is called the associated shift function so it is a function which actually provides its best approximation to a shift and this function preserves finite limits then concerning the categorical structure present on this category we have all small limits just because the shift condition can be expressed in terms of limits and limits commute with limits so actually limits can be computed just in the category of pre-shifts and in fact one can immediately see that the limit of the diagram of shifts will be again a shift in particular the terminal shift, the terminal object of the category of shifts is simply the function which sends any object of the category ok, now what about colinits as in the topological setting colinits are first computed in the category of pre-shifts and then you apply the associated shift function to the result and this will give colinits in the category of shifts just because when you start with a shift and you shiftify it you still get the same function so what you have is here the inclusion then the associated shift function and you have that the identity function that is isomorphic to that composition and so since the associated shift function preserves all colinits because it has a right adjoint we know that colinits are computed in this way another nice property that these categories of shifts have is the existence of exponentials for any pair of objects in fact exponentials are calculated as in the category of pre-shifts and in fact if you want to understand how to construct an exponential in a category of pre-shifts you just have to use the universal property of exponentials together with the unity of lemma ok, another important aspect of these categories of shifts is the existence of a sub-object classifier so what is a sub-object classifier well by definition it is an object usually denoted by ω which transifies the sub-objects in the category so the property is this so the object ω is characterized by the property that the arrows from a given object a arbitrary to ω are in natural bijective correspondence with the sub-objects of a in the category so this is called the sub-object so in fact the sub-object classifier can be concretely constructed in any category of shifts by taking the j-closed shifts on the category ok, then another crucial property of these categories of shifts is the fact that actually they have generators so or what I've written there separating sets of objects so what is a separating set of objects it is a collection of objects so you have e and you say that c which is a sub-class of the collection of objects of a is separating if for any a collection of arrows objects of c to a is epimorphic so it means that if I wanted to understand when two arrows between objects in the toposariquan it suffices to consider their composition with arbitrary arrows from objects in c separating set to a now can we find a canonical choice of a separating set of objects for a collection of topos well there is certainly a natural choice you can make if your topos is presented as the category of shifts on a certain site then of course you have a canonical factor which goes from the category to the topos so first you have the unit embedding which goes from c to the pre-shift topos and then you have the associated shift factor so if you consider this composite factor this is actually called the canonical factor sending the category to the topos and actually by the yonida lenma you can immediately see that the objects coming from the category under this canonical factor form a separating set for the topos outside or separating sets of objects is quite similar because in fact conversely if you have a separating set of objects for a topos you can keep this with a certain topology which will be induced by the canonical topology on the topos and you will get a set of presentation for your topos in fact in this way you don't get all sites you get just a wide class of sites which are called sub-canonical sites but still this shows that there is a very strict relationship between sites and separating sets in fact the solution that I can give you for understanding this is think about a topological space and a notion of base for this space so in fact you see you can have in general many different bases for a given space but each of these bases is generating the space and in fact when you go at the toposporitically level you see that the category of shifts on a space is equivalent to the category of shifts on any of its bases with respect to the induced topology so all of this is just a categorification of that already this shows that you can have completely different sites of definition for the same topos you see that there can be in general bases different bases for the space which are not equivalent that you cannot even relate directly to each other but which present the same space and which therefore will be rise to the same category of shifts but we shall come back to that when we illustrate the toposporitic approach to stone type so now let's go on by introducing the fundamental notion of morphism between toposis this is the geometrically motivated notion because of course we would like every continuous map between topological spaces to induce a morphism between the associated toposive shifts on these bases so by thinking about what we get starting from a continuous map we arrive at this notion of geometric morphism between toposis so a geometric morphism is of adjoint factors the right adjoint is called the direct image satisfying the property that the inverse image preserves finite limits then of course you can consider given two toposis and geometric morphisms between them a notion of geometric transformation between them which is just taken to be a natural transformation between the inverse image factors then we can define the notion of morphism of toposis the notion of point of the topos as a geometric morphism from the topos of sets to the given topos in fact this is a good definition because in fact any continuous map between topological spaces actually give rise to a geometric morphism well the direct image found is defined well in the obvious way you see this is the formula so you just consider the action on the open sets by the continuous map and then you decide that the value of the direct image of a shift on a given open set is the value of the shift at the inverse image open set then there are various ways you can use to describe the inverse image of the morphism but one appealing way is to use the correspondence between shields on a space and the etal bundles on this localomia morphisms so in these terms the inverse image factors can be described as the pullback operation of etal bundles along the given continuous map ok so we see that in fact any continuous map gives rise to an adjunction and in fact you can show that the direct image found preserves finite limits so we did get a geometric morphism and in fact if your spaces are sober every geometric morphism between the associated shift-opposites comes from a unique continuous map between the spaces so you see that really it is the good notion then we have that every growth in diktopos just by the fact that it is defined as a topos of shifts on sets admits a structure morphism to the topos of sets whose direct image is the so-called global sections found which takes the arose from the terminal object of the topos to an arbitrary object and the left adjoint to that is the factor which sends a reset to the coproduct of the terminal object indexed over the elements of this set then of course we have another fundamental example of geometric morphism which is the pair of adjoint factors given by the inclusion of shifts into pre-shifts and the associated shift factor which goes in the other direction this is a geometric morphism because in fact one can show that the associated shift factor always preserves finite limits now there is a very important result relating geometric morphisms because one in practice is interested in understanding geometric morphisms to a given topos of shifts so one wants to understand in a possibly more concrete way the morphisms going from a given topos to a topos of shifts on a site and well you see here we have a canonical morphism it is a geometric morphism from this topos to this so we can try to understand first the geometric morphisms to every shift topos and then building on that we will try to characterize those which factor through such a canonical inclusion so we will also get a description of those starting from the description in the pre-shift case ok so to understand this geometric morphism so we have to recall that whenever you have a functor A from a small category C to a locally small co-complete category E this functor induces a sort of home tensor adjunction between the category E and the pre-shift topos on C in the following way so we have a generalized home functor given by this functor which I have called rA and this functor admits a left adjoint which can be seen as a generalized tensor product in fact it is defined as the unique preserving functor which makes this diagram co-complete up to isomorphism so you can see this and actually the idea to understand geometric morphisms to a pre-shift topos is to observe the following that so if you have a geometric morphism like this which goes in this direction and in the inverse image we have to preserve arbitrary co-limits because it has a right adjoint namely the direct image and since this is the free co-limit completion of C actually the behavior of this functor is uniquely determined by its composite weave that you need in painting so this is the idea behind the representation of geometric morphisms in terms of suitable functors from the category to E so this functor will be given simply by the composite of the inverse image then the point is how to characterize these functors because you see this functor is not just characterized by the fact that it has a right adjoint and so you see if you start from an arbitrary functor then you will get yes an adjunction but this adjunction will give a geometric morphism only if this left adjoint preserves finite limits and so this leads us to introduce the notion of flat functor so we say that the functor is flat if this left adjoint preserves finite limits so as you can expect we have an equivalence between on the one hand the category of geometric morphisms to the free sheet topos and on the other hand the category of these functors so this is of course quite useful because it makes the description of geometric morphisms quite concrete after all I mean you can understand them just by seeing how they act on the representables now we look at what happens in the Schiffer case well in the Schiffer case you will just have to understand which conditions you have to put on the functor for the corresponding geometric morphism to factor through this and you understand that the condition you have to put is this j continuity condition which means the following a functor is said to be j continuous if it sends covalences to evimorphic families and so in this way you get you get an equivalence this is a very fundamental result which has a lot of applications in particular in connection with the classifying toposes of which we shall talk extensively tomorrow so this is what I wanted to say but I would conclude today's lecture mentioning a concrete characterization of flatness that it is possible to have so in fact flat functors can be equivalently characterized as functors that are filtered in the sense of this definition well of course if the category in which they are defined has finite limits then a functor is flat if and only if it is a finite limit preserving so this definition of filtering becomes interesting in the non Cartesian case so suppose for instance that the target topos is the topos of sets then a functor is filtering if and only if its category of elements is filtered as a category so in fact the general notion of filtering functor to an arbitrary topos is just a reinterpretation of the filtering condition in the internal stack semantics of the topos so it is just a reformulation of this condition using the language of generalized elements in the topos so I mean if you understand what it means in the set theoretic case you can make perfect sense of this more general definition so finally I would like to mention that there are two standard ways to induce geometric morphisms between topos one covariant and another one contravariant so the idea is to consider functors between sites with suitable properties so that they can induce geometric morphisms between the associated categories of shapes so one considers in particular two classes of factors the morphisms of sites and the comorphisms of sites so the morphisms of sites will induce geometric morphisms in the opposite direction while comorphisms of sites will induce geometric morphisms in a covariant way so what is a morphism of sites well formally the natural way to define it is as a phanto such that it is composite with the canonical phanto for the codomain topos is flat and the j continues then of course one might want a more concrete characterization of this which one can achieve in full generality it is a bit burdensome to give so if you want to get this definition you can take for instance my paper on denseness conditions but for the moment we can limit ourselves to saying that in the Cartesian case so if our categories C and D have finite limits then a morphism of sites so f is a morphism of sites if and only if it preserves finite limits and it is covered preserving so we need also it to send j covering 6 to k covering 6 so we perfectly understand what happens in the Cartesian case in the general case it is a bit more subtle just because of the filtering property for general phanthos but still we have a concrete characterization of them then we have the notion of comorphism of sites which is a sort of dual notion so it is a phanthos which satisfies the covered reflecting property which means that if I take a covering seed in the codomain category on an object which comes from the phanthos it should be a covering seed in the base category whose image under the phanthos is contained into that seed so the basic theorem is this one so you see an immediate application of this is to see how the our continuous map between topological spaces induces the geometric morphism you go through the sites of open sets and you realize that of course these have finite limits so if you consider the canonical topology isn't that this is a morphism of sets so when you go at the top of the theoretical level you get a geometric morphism going in the other layer which is actually the geometric morphism induced so I stop for today so tomorrow we shall start by previewing the view of toposes as mathematical universes then we shall present the classifying toposes and finally we shall talk about the toposes as unifying 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