 I'm very honoured to be able to talk here at the birthplace of Topos's. In light of all the excellent talks, I feel my offering may be slightly more humble, but you know, hopefully be a conjunction of some ideas existing and a couple of new ones will be interesting to some people and they might feel like looking into this further. Okay, so I have some terminology and a disclaimer. There was references to universes and so forth earlier in the week. If you like, take two universes. You're inside V, things in U will be small, things in V will be moderate. Anything not in U will be large, anything not in V will be very large. I won't refer to this again because it's in the background and sometimes you can dispense with this totally, but if it makes you feel more comfortable, take it. Also, unless I say otherwise, all Topos's, and I'm dealing with elementary Topos's a lot, so I'll get into that in a bit, all Topos's co-complete with natural number object, but I don't want to say take a co-complete elementary Topos with a natural number object over and over again. Okay, so the disclaimer is terminology. So here's some terminology which is sort of maybe four or five years old, but not widely circulated. So it's helpful to keep in mind, especially when talking about the relation between sort of forcing techniques which are coming from set theory in particular ZFC set theory and set theory not using set of C. So material sets is where elements of sets have independent material existence. So the elements of a set in ZFC say C is optional or bounded Zermelo set theory with choice, optional again, those who know the history of this area, this one turns out a lot when talking about Topos theory. These are material sets. Their elements are sets too, which are objects of the theory. Material sets is where everything is, there's no good technical definition, but it's roughly where everything is invariant on isomorphism. You don't worry where you say take the rules and then you have to think, do I mean the power set of the natural numbers, the functions from the natural numbers for the natural numbers, you know, the set of all dedicated cuts such the blah blah blah blah blah, you know, it's just the real numbers. It is what it is, they're all isomorphic. I don't have to worry about the elements accidentally coinciding with elements of another set. So structural sets, of course, levier's elementary theory of the category of sets. The primordial example, maybe some variance on this and one that I'd like to advertise called sets, elements and relations or seer due to Mike Shulman whose name we've heard a couple of times already. And this is not published, but it's on the n-lab, yes, thank you. It is a structural set theory that is not a categorical set theory. So you often hear people talking categorical set theory versus whatever ZFC, it's a structural set theory where the axioms are purely on sets, how they behave, elements, how they behave, relations, how they behave, three sorted, it's about four axioms, so shorter than the ETCS list. You know, and after like two and a half axioms, you've got yourself most of a topos. It's really quick. So yes, material sets and structural sets. All right, so let's do some history. Yes? So I've heard that ETCS is slightly deeper than ZFCS. Yes, it's the same strength as bounded Zermelo with choice, but this paper that Ingo mentioned in his talk, Stacks, Semantics and Structural Comparison and Material Structural Set Theories, it gives a formulation in this extended internal language, which is, which recovers full ZFC, a property called autology, and you know, when you can show that formulas are actually classified by sub-objects and so forth, it's very nice. So I can say structural set theory and recover full ZFC strength is no issue with this. There are other people too, like McClarty and others who have come up with replacement axiom analogs for the ETCS. Okay, so a bit of history, but in an a historical sense. So consider the topos of sets and then consider the subcategory of injections and then the pre-order, sorry, the partial order reflection of this is the category of cardinals and the usual order relation, the strict order relation here. So this is, you know, there you go, cardinals, easy. Half of Georg Cantor's life in one line. So the important thing to note is that the power set restricts to sets and injections because the power set of a subset is a subset of the power of the set of the superset, and the universal property here, this restricts, I want to call this on this side, so everything here is classical. We'll get to some more general topos stuff later. And this here is the continuum function, and of course the continuum hypothesis was, what value does this take when you plug in the natural numbers? Is there sort of some A and these are strict inequalities here? So as we all know, Gero constructed a model of ZFC where there is no such A. Any such A that's in between here with potentially equality, it's either the natural numbers or its power set. And then Cohen came along and gave us forcing and he said, well, I can construct a set A, well, construct a new set of a new model of set theory and I build it explicitly with a set A in here with these two different. It's a very nice write up of this in Mordech and McLean's book from Category Theoretic point of view, but they have some nice historical comments and so forth and they say you can actually see the shadow of the topos theoretic argument in Cohen's paper. I'll get to that in just a moment. Because there's a more general question. This is a class size partial order and you have an endo function, what can this endo function do? Given that we can find different categories of sets where it may exclude, it may miss values, what can it do? So there's two cases. So let's say behavior, there's two cases. The regular case, so these are technical conditions, singular case, so these are conditions on cardinals. The regulars are the nice ones, like the natural numbers are regular, assuming choice, things like the real numbers have a regular cardinality. Other things is like ALF sub omega, if that means anything to you. Singular cardinals are sort of weird, this is hard and open. This is solved, or Eastern. We know exactly what the behavior of the continuum function is on regular cardinals, in the sense that we know, we can characterize precisely what behavior it can have, what it can't do, et cetera. So, yes, and people like Sheller, Sheller himself literally writes hundreds of papers on this, but other people write hundreds of papers generally, trying to attack this problem for the singular cardinal hypothesis and so on. So why is P so important? This is a set theory notion, why do we care about it as Topos Theorist? So here's a definition of an elementary Topos, because I don't believe we've had one, well, I think one was given in another lecture, he wasn't happy with it. Here's one I'm sure some people may like, finite limits, power objects. And we know a Topos has incredibly, incredibly rich internal logic, you can do all sorts of things, you know, code limits, limits, you know, descent of certain things. Finite limits is pretty weak, everything else comes from the power objects. In fact, you can relax this even further, but then you're starting to get silly. So this is saying, you know, for all objects, X, I have another object, PX, a relation from X to PX, such that any other object, Y, a relation from X to Y, there's a unique map, F, and such that this square is a pullback. So this is just, all the magic happens from this, it's just incredible. So looking at the behavior of the power object functor, or the power set functor in the set theory case, you know, somehow it has incredible freedom, as we see from the set theory, but in fact it gives us lots of structure, so it's good to sort of understand what's going on here. I mean, this is solved purely in classical logic with choice. If you throw away choice, set theorists run screaming from the room and say, I don't know what co-finality is, it doesn't make sense. Why do you want to do it? I don't know, maybe. So there's, alright, so forcing, I've got to talk about forcing. So forcing is chiefs, this is full stop, there's nothing mysterious about it, details perhaps, but here's the cartoon I'd like to draw. Start with ZFC, start with a small partial order, and it's a site with a double negation topology, take sheaves on it, ignoring that. Alright, now you have two options here. You can take internal logic, and then you just reason in that, it's perfectly good, and then you perform a construction due variously to coal, Mitchell, OCS, Forman and Hayashi, use facets of it, and then you get what the set theorists call a Boolean valued model of ZFC, and the Boolean is the topology on this partial order, take the Boolean algebra of regular opens, and then this Boolean algebra is where your truth values lie. Another way to do it is go further, take, let's see, partial order with top, I mean, yeah, post set, it's without loss of generality, assume it's got a top element, but that's, yes, any small post set. And what is the double negation sign for post set? So this is the site, this is a small site, double negation topology, sieves are downward closed dense sets, so there's an analog of this topology for arbitrary categories for the dense topology, it corresponds to the double negation topology when you pass to the pre-sheaves on that category in the sort of Vietzioni topology sense. So, yeah, partial order, think of it as a category where, like this, and you have a top element and you think of it as being going down, and the further down you go, it tells you more and more partial information about the things you get in here. Okay, so that's one thing to do, I don't know if you want this, or you take in your category of sets, the external sub-objects of the terminal object, and you take a filter, so this is a bit like a point, but not quite because you require different things on this fire, and you can perform what's called the filter equation construction. The details are not important, but this here is now well pointed. The internal logic and the external logic now agree in this category, and so you can reason exactly as if you were in the internal logic, but you just go, well, this is how the objects of this topos works, and then you perform this construction again in various of these various people, and then you get, you know, the forced model. So set theorist will sort of flick back and forwards between these two very points as convenient. Now the thing is these dotted lines, this is a rather delicate and complicated construction involving transfinite well-ordered rooted trees. Set theorist don't like it if you say that set theory is a study of well-founded pointed trees, but it's not really, but that's what they're studying. So this is the step that we like, and perhaps just tack on the internal logic, because that's where all the action happens. Yeah, and if you read Cohen's paper, you can see he's describing, you know, exactly these constructions in some sort of senses, like McLean mentioned. There are all sorts of variants. You can consider a topological group acting by automorphisms and my partial order, and then consider equivalent sheaves, this is called symmetric submodels. You just take a topological group, full stop, and consider continuous actions of this. This is what are called permutation models. They don't give you models of ZFC, but with atoms. So this predated Cohen by about 40 years, and it's basically forcing, but set theorist like that's not real set theory atoms. So, but it fits into the category theory picture perfectly well. Okay, here's the important point. Taking a poset can only affect set many, you know, the values of the continuum function. You know, if we start with a model of set theory, look at the continuum function, do some forcing, the continuum function changes, but this procedure with a poset can only affect set many. It's a particular bound above, based on the cardinality of P. The problem is Eastern solution involved adjusting a proper class of regular, the continuum function at a proper class of regular cardinals. So you can't sort of go, you know, literally you can take all the regular cardinals of which they're, you know, I can't say most of the universe, but it just keeps going up. He was able to shift them, you know, subject to the hypotheses of this theorem. No, no, that means a set. There's a set, and that is the cardinality of the number of values of the continuum function which you have altered. So you can't affect arbitrarily high values of the continuum function by taking sheaves on a small partial model. Okay, and what he did, he did, you know, he genuinely did it. He adjusted a proper class of values of the continuum function one at a time. This might make you pause. Discussions with us. Okay, question. Yes, please. Yes. Okay, speed it up. All right. So there's a notion, there's two notions from SGA which I found which are very beautiful. So there's notion of four topos. This is for growth index and so on. This is because they are thinking growth index topos. And so for us, this is an unbounded set topos. So this is a topos with a geometric morphism to set, but E is not a growth index topos. In particular, so the example they give is you take a large pro group and then you consider, so again, universes as needed, you take the limit of this pro group and then you consider continuous actions of this. And this is, you know, up to some small care as to what you mean. This is a perfectly good locally small category which is a topos with a geometric morphism to set. And, you know, it is perfectly good topos. And more over, if you write down naively a site of definition for this, it is indeed a large site. I have an application of this in paper in Studio Logica early this year. Actually, it's quite nice. And it uses this unbounded quantifiers internal logic. This is too restrictive. This is not what Eastern used. Consider, let's see, filtered and well founded category, initial object. You can consider a diagram of sites. And I'd like to advertise the morphisms which are vibrations of sites due to Mordak. You can take sheaves, go to topos with bounded morphisms. And this is an example, well, it gives rise to an example of a fibred topos, also an SGA4. And they consider the case where r is small. And you can take the total topos or the lax in particular case, probably want the pseudo limit of this. And they get a topos and people have apparently applied this in a direct geometry. But if r is large, you can't, you just can't do it. If you forget to, in fact, you probably want, sorry, bounded. These are growth index topos. This is sheaves. So this morphisms of sites, so this is probably what you'd call a morphism of sites in the growth index style. But in fact, this is just the right adjoint to a vibration. And this notion, if you think of sites as like a basis for a topos in this sort of sense, we've heard today, a vibration of sites is almost exactly analogous to Jordan normal form. I mean, it's not true. You can pick C, just pick a basis for one of your topos and then you pick an internal side of definition for the topos sitting over the sheaves on this. And then you can pull this down and you get a very nice expression. We have these adjoint functors with cover preserving properties. You can characterize, it's extending this site characterization to characterizations of geometric morphisms by these particular vibrations. Monarch covers a bunch of examples of these. It's just, I think it should be advertised more widely given this sort of viewpoint of sort of using a basis. Okay, so get back to here. What is it, CC? Co-continuous Lex. And this is, this is now a very large category. So I'm thinking of topos is I have sites and then she's over them, see it on here. Lex is sort of starting to creep up just one more universe. So what is the two category then? This is, this is normally means left except, but if you are talking about co-continuous, then you are calling it probably as well. Yes, because growth in topos is co-complete and I'm taking the inverse image function here and I take the two arrows which make the conventions work. We've seen there are different conventions, you pick the one that works. So what is the Lex, what is Lex? Finite limits, co-limits. I mean it's what you might call a presentable category, but minus the actual presentability. Because these are, they factor through presentable. This sort of algebraic notion, but I want to go to the Lex because what happens next is I take, so let's call this big diagram E and I'll let E star be opposite. So R to Lex CC, R is a filtered, well-founded category with, think of it as the ordinals with, sorry, initial object. Initial object, not a zero object. Initial object, and then you have a map from this to, to sides with vibrations. Yeah, and then you can take topos and then you forget, you just take inverse images, the inverse image part. These are like the algebraic viewpoint on topos. I'm forgetting the direct image functions because the direct image functions will no longer exist in the next step. And I take the co-limit over R of E star from R. Now this is not a locally small category in general, but you know you have some universes or you take the first order theory of categories and this is what Eastern does. He picks an appropriate R, in fact it's just a class of regular cardinals. He defines appropriate sites which you know affect one regular, one value of the continuum function at a time and then he takes a co-limit. It's affecting all regular cardinals sitting in my diagram category. That's a two-kategorical co-limit. Yes, that's pseudo. So sometimes I call this the Eastern, it's not a topos, a priori. So you've got to say the theorem. So I mean he has very specific choices, yes? So E is, it's almost a topos, co-complete, hiding pre-topos with finitary, finitary w-types, e.g. a natural number object. So this is the internal logic here is first order intuitionistic logic. We have hiding algebras as your sub-object lattices. W-types means it has algebras for initial algebras for polynomial functors associated to maps which are numerals in a slice category. So very, you know, literally a finite polynomial. Natural numbers are an example of this. And if, if, so notation is above, so, so for every indexing object, every topos, every object in it, and I say power set of x is eventually constant, and you can formulate this in a, in a, just a straightforward technical sense. It means as you keep going by your inverse image functors, eventually the power object stabilizes and then is isomorphisms all the way up. It doesn't change. Hey, oh what for a moment? If this is true, then he is a topos. This is exactly the condition that Eastern finds and characterizes the special case of vibration of sites in that he's using that makes this work. And he's doing localic topos, so all he needs is not for every object in the indexing category, but just in the base category. So if every set in my base category, if its power set eventually stabilizes after passing by the inverse image functors after this class long diagram, then he gets a topos. So you get power, so adding power objects, so all you miss are this. So the sequence of the power objects of the representables has to be used as the stabilizer? Yes, yes, exactly. And in fact you don't need all of them, you can just take some co-finite sort of collection inside the, inside the order of cardinals. For large enough, you don't need to check all of them, you just need an unbounded sequence of cardinals. So here you don't have internal, you don't have internal homes, but adding a power set you get all the topos set structures, but I think I'm over time, so I was going to give, explain Eastern's example of how he characterizes this eventually constant in terms of properties of the posets, but you can see me if you're interested. Thank you very much for your time. What was the, can you recall what was the notion of vibration of side? Ah yes, yes. Let me just, so P is a vibration of categories, so you know whatever Cartesian lifts so on. It is left adjoint to a fully faithful functor going in the other direction and T preserves covers. You're speaking now about sites or topos? These are just sites, these are just sites. So I didn't write the topologies here, but this is the underlying categories and yeah these are, there are topologies here in here. And there are no special assumptions like if this is a finite limit? No. But without loss of generality, if you have complete freedom you can actually choose C and D to have finite limits if you like. But there are important cases when they don't have them, so it works in more generality. Because the notion of reserving coverings is not as good, so if you don't find it. Yeah, it's like flatness or something really. Yeah, yeah, yeah, yeah, yeah, yeah, the image of a covering sieve generates a sieve and you know various things here. So it's in Mordak's, it's in the elephant and Mordak's paper from I should say what year it is to site, site, site, site, site. 1986. So this is a very nice notion and it's only in his paper and in the elephant, and the elephant doesn't elaborate on his paper. So I think it should be pushed more because it's quite a useful technique. So is the category well-powered the one on the left, like what that you get when you're hiding? Here? If it's a co-complete hiding grid topos, you're not okay. So because I was about to say that the hiding is automatic if you're a co-complete topos. So it's sort of like small, sort of small hiding if you like. Yeah, and it's not even well-powered once it becomes a topos, but it has power objects and it's because of this lack of local smallness that you don't get the adjoint functor theorem to get all the direct image functors that you might think from the co-cone of the diagram. So this is some like a limit except it's really only in the algebraic morphism of topos sense. You can't find the right adjoints. Bounded morphisms. But for all morphisms, if you use your Schultz-Groentz-Tropos sense. Yeah, yeah, yeah, but that's over set. I mean that's like topos slash set. Groentz-Tropos is over set, then automatically morphisms between them are bounded, but I have to forget set anyway, because I no longer have the right adjoint, the global section is functed down to set. So I have to forget that I'm sliced over set. This is definitely because of the size of the issues. Yeah, so I probably have something like up here instead of like top over set, as you know, Groentz-Tropos is, I have lex categories under set. And so I have constant sheaves that are in this big, big topos curly E, but I don't have global sections. Yes, when you say co-completes the topos, do you mean what Peter Justin called a infinity pre-topos or is there a problem with that? Oh yeah, yes, no, sorry, yeah, it's an infinitary pre-topos, sorry. So they call it an infinity pre-topos, but we have infinity topos. So yeah, co-limb, all co-products, small co-products. Yeah, the co-compolises you'll get for free because you have a natural number. Yeah, you can generate equivalence relations.