 So the last talk this afternoon will be by Jörg Bidermann and I mean he will talk about higher shifts. So Jörg, it is yours. No, you can go. Thank you very much. I would like to thank the organizers for this nice meeting and for inviting me. This is joint work with Mathieu Annel, Eric Finster and André Géal and the results that I'm going to talk about in these in these preprint higher shifts and left exact localizations in infinity top it's available in the archive. Some related results are in these other article modalities in homotopy type 3 by Reike Schulmann and Spitters. Okay, so let me start by recalling some facts about one top way. So if you have a small category C and you take pre-shifts with values and sets, that's a one topos. And every one topos is the left exact localization of such a pre-shift topos. And then it's true that any left exact localization of a pre-shift topos corresponds bijectively to a Grotendig topology on C. So any left exact localization is given as the sheafification with respect to this Grotendig topology. And then there is also this set theoretic thing that in in one topos any left exact localization is accessible. Okay, now I'm going to infinity topoid. So when I saw C is now an infinity one category and usually I will just say category and S is my notation for the category infinity group points and I will basically just say the category of spaces and an object I will call a space. It's modeled by topological spaces up to weak equivalences or by simplificial sets, for example, up to weak equivalences. And then again we have the basic fact in infinity topos theory that if we take a small category C and then we take pre-shifts on that with values in the category of spaces, well that's an infinity topos. Okay, and then it's still true that every infinity topos is the left exact localization of such a pre-shift topos. Okay, here I have to say accessible. There might be things that are inaccessible but I let them be inaccessible. But then here comes the catch now in higher topos theory not every left exact localization of such a pre-shift topos is the sheafification with respect to a Quotenic topology. So Quotenic topologies give us left exact localizations but there are more. Now how about that? So we are faced with a few questions. So given a small category how do we describe all the left exact localizations of a pre-shift topos? Or more generally if we are given an infinity topos E how do we describe all the left exact localizations of E? Or slightly reformulated if I'm given a set S of maps in an infinity topos? How can I invert S in a left exact way? So I would like to describe the left exact localization generated by S. I know how or we know for a long time how to localize with respect to S but it's not clear how to left exact localize with respect to S. And then finally we need to say what is a sheaf with respect to S. And we can answer all these questions. So the answer will come in a few steps. I will define a nested sequence of full subcategories of E. So given a set S of maps in an infinity topos E I will first define the local objects with respect to S. Then the modal objects with respect to S and finally the sheaves with respect to S. And I will try to explain how all these works. So first so we are given a set of maps S in an infinity topos E and then I call an object X in E. I call it S local if for any map F in that set S pre-composing with this map induces a weak equivalence of these mapping spaces. So recall that in an infinity category an infinity category always comes enriched over the category of spaces. So this map E is the mapping space. It's a space. And I'm pre-composing here with F and if this is a weak equivalence for all the F's in S then I call X S local. This is an instance of orthogonality that Charles Ress mentioned in his talk. So we basically this is the condition saying that the map from X to the terminal object is right orthogonal to all the maps in S. Okay and I will write log ES for the full subcategory of S local objects in E. And it's a fact that this log ES is a presentable category and it's reflective in E. So reflective means that the inclusion function has a left a joint and this left a joint is the reflector but sometimes I will just call it the localization slightly imprecisely. Okay next I will tell you what are the modal objects. So an object X in an infinity topos is called S modal. If it's local with respect to all the maps in S and all their base changes. So that's us now a bigger a bigger class of maps and so so modal objects are local with with respect to they are local with respect to a slightly bigger or bigger class of maps. And I will say I will write mod ES for the full subcategory of S modal objects. And now it's true again that these modal objects are a presentable category and they are reflective in E. So why is that? Well there is a trick in the definition of an infinity topos one always says that E is a presentable category. So presentable means that there's a choice of generators for that category. And if G is such a set of generators so there's a choice of set of a generator of generators and if G is a set of generators for E then in the above definition I don't need to take all the base changes of the maps in S but only the base changes over the generators. So I'm taking my maps in S and if the target of this of this map in S is B then I take all the maps from the objects in G to S and I pull back along them that gives me now a new set because S is a set and G is a set. So now I have a new set which is bigger but it's still a set and then the modal objects are the local objects with respect to this bigger set. Okay so I need to now add base changes. Next I need to define what is the diagonal of a map. So if F is a map from A to B then I pull it back again along itself and then I have a little square and the diagonal there is some other co- Cartesian the Cartesian gap map of this of this little of this little square. And then I need to define iterates of the diagonal so the convention is that delta zero of F is F and then delta N of F is just the diagonal of the diagonal of the diagonal and so on. Example if I take the map from A to the terminal object and the diagonal of that map is just the diagonal of A A to A times A. And the nth diagonal of that map is A mapping to A to the S N minus one. So for example in spaces this A to the S N minus one is just the functions from the sphere to A and among them you have the constant functions and this A that maps they picks out exactly the constant functions. And then I can define the diagonal closure of this set S so I take all the the maps in S and then I add all their all their higher diagonals their iterated diagonals and I call this set delta infinity S and I will call it the diagonal closure. So now I can give you our definition of sheaves. So S is a set of maps in an infinity topos and X in E is called an S sheave if it is modelled with respect to this diagonal closure of S. And the full subcategory of E given by the S sheaves is denoted by sheaves ES. And our main theorem is the following the category of sheaves ES is presentable and reflective in E and the reflector that goes from from E to the sheaves in E is a left exact localization. In particular the sheaves themselves form again an infinity topos and L has the correct universal property that we would like it to have. It's initial among all the left exact and co-continuous functions that invert the maps in S. So left exact means it preserves finite limits and co-continuous means it commutes with all co-limits. So in this way we this is the left exact localization generated by my set S of maps. And now I can define what is a higher site. So if I'm back in the case of a pre-sheave topos I take as generators the representable functions and I call RC the set of representable functions in this pre-sheave topos and then an infinity site is just a pair of a category C with a set S of maps in the in the pre-sheave topos. And then a sheave with respect to this infinity site is just a pre-sheave which is local with respect to all the RC base changes of the diagonal closure of S. So this is the recipe to get to get sheaves. You take first your set S you add all the higher diagonals then you pull them back over your your generators in this case the representable functions and then you take the local objects. Now this definition is not exactly the definition that that Charles gave in his in his lecture. In his lecture he defined a higher site but S in his case was just a set of monomorphisms. But here we have a general set of of maps and we can define now the sheaves with respect to a general set of maps. And then we have the corollary that any infinity topos is equivalent to an infinity topos of sheaves of one such an infinity site. Okay I would like to to now relate this notion of sheave to the classical notion of sheave. For that I need the notion of a monomorphism. So a monomorphism is a map such that it's it's diagonal is an isomorphism. An isomorphism here is to interpret it in the higher categorical context so you would probably say a weak equivalence. Now a monomorphism is the same as a minus one truncated map. It's just a it's just another name for that. And in the category of spaces a monomorphism is just the inclusion of a of a bunch of connected components into your space. And in the category of sets a monomorphism is just an injective map. And there's this definition by by Jacob Lurie. So if your left exact localization is generated by a set s and this set s only consists of monomorphisms then you call this localization topological. So topological localization is a left exact localization that is generated by some monomorphisms. And then Charles Resk explained he sketched somehow why these localizations these topological localizations of a pre-sheave topos correspond exactly to the golden topologies on C. So these are the ones that that we already know somehow. And so what is now that the classical sheave condition is just somehow if you take a set of of of maps in an infinity topos that only consists of monomorphisms then you don't then somehow the first step that we took is to take the the diagonal closure of your set s. But if you have monomorphisms well then the diagram is an isomorphism and the higher diagonals are also isomorphisms. So what you add is just isomorphisms you have your set s plus a bunch of isomorphisms. So those ones you don't need to invert anymore they are already isomorphisms. So this means that in this case the sheaves are exactly the same as the modal objects. The step of adding diagonals was unnecessary and then our sheave condition reduces exactly to the classical sheave condition given by golden topologies. Because now you have a set of of monomorphisms and when you take this this step where you pull back over the generators well if you pull back a monomorphism you end up with the monomorphisms in it. Now you have monomorphisms with target the representable fronters this is what what people call a sieve right this thing is now stable by base change because we just added all the base changes. So this is how you get a classical sieve and the proof I just explained here if you have a monomorphism then the diagonal is an isomorphism. Now in one topos theory this is all there is every leg's localization is topological so why is that? Well if we take a map f from a to b in a one topos and we would like to invert it in a left exact way how do we do it? Well we factor it into a suggestion followed by a monomorphism and then I start inverting the monomorphism h here in the factorization and all its base changes. I need to add the base changes in order to make it left exact the localization but that's fine so far we are only inverting monomorphisms and then in the next step I need to invert the diagonal of f well the the image of the diagonal of f is the diagonal of the image because the localization induced by inverting h is is is left exact so now I invert the the diagonal and and this makes then the the the map f also a mono so it's a suggestion and a mono and therefore it's an isomorphism so this is the the recipe to to invert a map in a left exact way but now comes the point in our one category the diagonal of any map is already a monomorphism so if you do the next step the second diagonal it's an isomorphism there is nothing anymore this this is all there is you you get away in a one-topos to generate left exact localizations in a one-topos you get away by inverting only monomorphisms therefore they are all topological in higher-topos theory that's not the case if you have a space and you take the map to the terminal object then the diagonal of that map is the diagonal of a so you go from a to a times a and again in spaces yeah you have or in sets you take you you can draw the the square and then you can draw in the diagonal it's a subset but in in the higher category of context this is not a sub-object if it were a sub-object all the fibers would vanish let's compute the fibers of this map well in the higher category of context I need to compute not the the strict fibers I need to compute the homotopy fibers and how do I do that I need to replace my map my diagonal with a vibration and one way of doing it that is by replacing a by the path space so now I have maps from the unit interval to a and it has two loose ends the unit interval and I can evaluate at each one and this gives me a map downstairs to a times a this is a vibration and this is the one that replaces my diagonal and then I calculate the fiber so the general fiber at the point a comma b in the in the product is now the maps or sorry the paths that started a and ended b now if a is equal to b so if I take a point in the diagonal in well then the fiber is a loop space the loop space of a at the point a and in general in homotopy theory they don't vanish that it's just a fact of life they get into the way yeah and so the diagonal of a space is not a monomorphism you cannot do the trick I just did in in one category here you just you have to add higher diagonals so let me just see how many how much time ah I'm very fast I'm very fast okay so so what is left what is left these are the luri recognized this and he called them the cotopological localizations they are the ones the left exact localizations that invert no monomorphism at all and he also proved the theorem that any left exact localization in a in a topos in an infinity topos can be factored into a topological localization followed by a cotopological localization and that well the topological localizations are the ones we somehow we know I mean for a long time people know how how to deal and to handle uh gotten the topologies and sheeps and things like that but what what are these cotopological localizations I mean maybe they are just an artifact of of higher topo theory what about them well in fact they are very important somehow I'll give you an example let Fin denote the category of finite spaces so these are the spaces whose homotopy type is equivalent to a finite cw complex and then I take the the category of functors covariant functors from finite spaces to spaces I so we denote them by s a joint x because it has the universal property of a of a polynomial ring but now our ground ring is the category of spaces somehow it's that the infinity topos generated by by one object and uh in this in this infinity topos I can take evaluation at the terminal space or the one point space and this gives me a left exact localization well it also preserves all limits because evaluation preserves all limits but I that's not so important at the moment I I take evaluation at the point and I call it p zero p zero why because it's the zero level of the good willy tower the good willy tower is supposed to be a taylor replacement of the taylor series and the local objects with respect to p zero are the constant functors so like for any taylor series for any good taylor series the the zero level is given by the constant things and this localization p zero it has a a non-trivial topological part but it also has a non-trivial cotopological part and uh somehow this is somehow what what forced us to consider cotopological localizations because in our in our group the four of us we started because we wanted to consider to understand the relation between good willy calculus and higher topos theory and then this observation it forced us somehow to come to grips with cotopological localization and then we started to understand how to generate left exact localizations and everything and then somehow the point is that's in a forthcoming paper is that good willy calculus happens completely on the cotopological site somehow so the fact that these cotopological localizations exist gives rise to the good willy tower and the good willy tower even though you might know not know what it is doesn't matter it's central to homotopy theory so this this is supposed to tell you that cotopological localizations are not something pathological they are in fact central and uh let me just see maybe i have time maybe i don't explain this well okay this this somehow um this slide uh illustrates how our theorem works so if we want to generate this localization p zero according to our theorem we can do it in the following way we take x well this x it can be identified with the function the inclusion of finite spaces to two spaces so it's the identity function and then i consider the map from x to the terminal functor the functor that is constant the point and this so these two functors they are representable the the identity is representable at the terminal object and the terminal functor is representable by the empty space and now we can we can go through the list what are the local the modal objects and the sheaves well the local objects are just the ones that such that the map from the initial object to the terminal object are isomorphic okay because you see if you if you map this this map from x to the point to f you use the yoneda lemma and then you get this this this map from f to the to the of the empty set to f to the point and being local forces you that this map is a is a is a weak equivalence okay well that's maybe not so interesting what are the modal objects well now you can add base changes so a base change of the map upstairs looks like this x times a representable functor to a representable functor the representable functors are now covariant and a modal object now is just a functor such that the value at k is the same as the value at k union at this joint point okay that's maybe also not so interesting but now what are the sheaves well now recall that in fact the first step to to generate sheaves is to add all the higher diagonals well the diagonals they look like this x to x to the sn minus one and x to the sn minus one again is a representable functor it's represented representable in a sphere and now when you take base changes of those maps you will realize that they give you they are the yoneda image of of of maps from from a sphere to k and the cofiber is gluing on a cell it's gluing on an n cell and any finite space is obtained in a finite wave from from gluing on these cells so now a sheave an x sheave is a is a functor that sends that that doesn't distinguish between gluing on sheaves so f of k and f of k with the gluing on cells f of k with f of k glued on that n cell is just the same so this means that all the values are the same so the the f is now a constant functor so this means that p zero is generated by this single map from x to the point and now our theorem is that the good willy tower is somehow just generated by taking higher joints of of that object x somehow what you do is you take you take the somehow the kernel of your we call it a congruence class so that's the map that i inverted by your localization in this case p zero and then you take higher powers of this thing which is an ideal you should think of this like an ideal of a ring so now you take higher powers of that ideal and you divide it out and what you get is the good willy tower in fact thank you very much that's all a lot okay i mean you have you have really explained quite well so are there questions from the audience i mean i guess i have a you know general question more philosophical i mean i understand very well that you want to pass to spaces and so on and so forth but i mean what is really the origin of infinity topos is i mean in that in that i mean i mean in the very general broad philosophical sense i mean well the fact is that that somehow these functors from finite spaces to spaces it's it's just an infinity topos right you mean this is this is a basic example you mean that i mean for for me i mean the basic example is maybe the category of spaces itself that's an infinity okay and then let me say it like this so but i i'm sorry i mean when you talk about the category of spaces you mean for instance semplicial sets or what i mean yeah i mean i mean the well i mean the infinity category associated to for example semplicial sets yeah okay okay yeah as an infinity category okay yes via kan localization sure okay fine okay and uh so spaces and uh infinity let's let's say co-complete infinity categories you can think of them as the the modules over spaces so spaces are now thought of as a ring yeah okay fine fine that's perfectly good yes yeah and the co-continuous sorry the co-complete infinity categories are now like the modules okay and the infinity topoi are like s algebras they are like the algebras in the world and then somehow but yes wait wait a second i mean i mean if i if i take the analogy with gamma sets uh where i don't have spaces but they have just sets i mean okay yeah so um gamma gamma gamma gamma spaces so they're that the source category is a is a is a one category right it's just pointed pointed sets or the opposite of pointed sets out just points finance yes let's say yeah yes and uh in this case you don't have cotopological localizations no no of course no yeah so they are still of course topological localizations somehow but uh what what we want is we we want uh we want uh finite space so if you if you take functors covariant functors from finite pointed sets to spaces you can left karn extend them to all spaces exactly yes yes and then you get functors that are they are called um how do you call them maybe strongly finite area they commute with all shifted columns yes that's that's it's a nice class of functors but it's it's a bit restrictive when you take finite spaces as your source category and you do the left karn extend these are now finite area functors these are the ones that commute with filtered columns not with all shift columns only with the filtered columns and this is somehow the place where where good willy calculus happens and then you can think of that really just as as a as a ring yeah yeah okay as the analog of the ring yes yeah and and here and somehow we we we can give we can give this interpretation in fact we have a more general construction which we call the nilpotas tower which we are still about it's in the forthcoming paper it's not yet on the archive so you take any infinity topos and any left exact localization and to this data you can associate a tower of left exact localization which is like the good willy tower which is like the good willy tower exactly okay and the good willy tower is an example as is the orthogonal tower by vice and there's also a unitary tower by nil there are all examples of that oh that's really good yeah okay so thank you and bye bye thank you very much