 Now for something completely different, categories of spaces that are not categories, so categories of spaces that are not categories of locales. My plan of attack is I'm just going to recall some properties of the category of locales, notably that there is a double exponential object in that category. Then I'm going to look at that double exponential object in any category and prove a stability result for that double exponential object. And that will actually give me a class of categories that has this double exponential ability to it. And then the sort of point of the talk is to show that that class, what an example of a category of that form with a double exponential ability cannot be a category of locales. And then if I've got time, I'll put a little bit of sort of meat around the bone and sort of justify saying actually a category that does have this double exponential ability is like a category of spaces. So you can do a lot of topological space or locale theory within that category that satisfies those axioms. So I'm going to be talking about lock category of locales. I'm going to write this for the Ciepinski local, Ciepinski local. And the main property that we have is that for all for all locales x, Ciepinski to the Ciepinski to the x exists. Now Ciepinski to the x doesn't always exist because not everything is locally compact. So what do I mean by that? Well, we do have a pre-chiffunctor, Ciepinski to the x going from lock up to set. So from the opposite of the category of locales to set. And that sends any y to lock y times x comma Ciepinski. So you can define that. And that is the exponential in the pre-chiff category. So when I say that this exists, what I mean is that your nadir of Ciepinski to the power Ciepinski to the x, in other words that pre-chiff, exists in the pre-chiff category lock up comma set and is representable. And when we have that, we write p of x for the double exponential. And it's a property of the category of locales that this double exponential exists. And the key universal characterization, the points of px unwinding the double exponential is that they correspond to natural transformations morphisms in the pre-chiff category from Ciepinski to the x, Ciepinski to the y. And once you have that, you have that p defines a strong monad. And it's also interesting to note that once you think about this pre-chiff category, you can do actually quite a bit of topology. So for example, you can prove that a locale map, f goes from x to y, is open if and only if there exists a left adjoint to Ciepinski to the f. So those are natural transformations in the pre-chiff category, such that the usual Frobenius reciprocity condition holds. And it's also well known that once we can say what an open map is relative to any category, I guess with finite limits or these finite products, we can say what the discrete objects are. So we say that x, in any context where you've got some objects of Ciepinski, x is discrete if and only if the finite diagonals are open. And all of this works relative to an arbitrary elementary topos. So you can have the discrete objects relative to loc, relative to any e, or that's just e again, and it embeds in the category of locales relative to e for any elementary topos. So the theorem that I want to prove, or at least outline the proof, about double exponential ability is essentially saying that double exponential ability is closed under the formation of G objects. So we've got a proposition. So C and it, C just has to have finite products. That's my only assumption on C. So hopefully if nobody's interested in any of the locale theory we'll talk about spaces, they'll see this as an interesting result about double exponential ability. So you have an object A such that A to the A to the X exists in C for all objects X. We're going to have G, which is a group, excuse me, which is a group in C. So it's got products so we can talk about the internal group. Then the result is that A to the Pi 2, A to the Pi 2 X, A exists for all G objects X, A. And just the clarity, so I'm talking relative now to this category. Just use a pre-sheaf notation for it. It has as objects, it has G objects as objects. So a typical example is X and G times X goes to X. And of course the Pi 2 is just A with the trivial action on it. So can you just read the indices again? Sure, this one here. So that's A to the Pi 2, so A with the trivial action on it, A to the Pi 2, A with the trivial action on it, X, X comma A. So for an arbitrary G object this double X exponential exists. So just to go through the proof of that, or at least in outline. So I need to construct this object here relative to G objects. So P of G X comma A is going to be, have as its objects the double X exponential with respect to X. And have as its G action something that's defined through the strength. So the strength of the monad, once you have double X differentiation you have a strong monad. So you have a strength morphism here. That goes from there and then it sort of shouldn't surprise you that it's P of A is how the action on P of X is defined. So to complete the proof we need to show that, let's get that right, we need to show that the collection of G homomorphisms, let me just state that, so to complete the proof we need to say that for all other G homomorphisms say Y comma B that the collection of all morphisms from Y to P of X that are actually G homomorphisms that these are isomorphic to the natural transformations from now A to the Pi 2 to the X comma A, A to the Pi 2, Y comma B because that's exactly the universal characterization of the double X differentiation that I gave you earlier. So in summary at least, this left hand side, well these correspond to natural transformations so they correspond to alpha that goes from Ciepinski to the X, Ciepinski to the Y and then unravelling the definition of G hom in the pre-shift category gives you a particular square, Ciepinski to the G times X, Ciepinski to the G times Y, it's almost sort of what it has to be, Ciepinski to the B, Ciepinski to the A, I'm sorry I realised that I've done a notational error so you'll have to bear with me but my Ciepinski is equal to the A so where have I made that error? I'll just replace it here for good form that's A to the X, so unravelling the left hand side in terms of the construction of that point in terms of natural transformation you have that, that G hom condition you sort of has to be that you need to unravel the definition of strength. On the right hand side here I'm going to apply the following co-equaliser in the pre-shift category so this actually embeds in an equaliser diagram here, right I'll just say what it is, so that's G times Y, the free the free G algebra on Y, that's G times G times Y again the free one and the reason why this is a equaliser in the pre-shift category is that effectively you can apply the exponential to the well-known expression of the algebra Y comma B as a quotient so there's a in the category of G objects we always have this quotient and it's a split we always have this quotient in the category of G objects and if you apply G to that quotient to that co-equaliser you get an equaliser over here because you have an equaliser over here these natural transformations embed into a natural transformation from A comma Pi 2 to the X comma A, A comma Pi 2 now it's G times Y comma M times the times the the identity of the free sorry the free action on Y so you just got M times Z so this embeds in that because of that co-equaliser and the final step in the proof is to observe that this is what I meant by free you've got this left a joint you've got to forget from there that's actually monadic because that's a category of monads over the monad induced by that but it also satisfies Frobenius reciprocity and because it satisfies Frobenius reciprocity this actually extends contravariantly to an injunction in the in the pre-shift categories um so that's comma blank over there so unwinding that injunction in the pre-shift categories we see that these natural transformations actually can be related back to natural transformations from A to the X comma A to the Y because this is the this is the free G times Y so free is on the left left a joint it's now contravariant so that's down on the right a joint and it's gone down and there's a little bit of checking to do that the condition here corresponds to equalising with those two arrows so there's some additional check that I'm not going to go into but that's how you show that if you have an object in an arbitrary an arbitrary category which just has finite products then double exponential ability is is G stable right so the now I want to sort of make good the sort of promise of the talk in terms of giving you an example of a category that has this property but is not necessarily a category of locales so I'm going to prove so I'm going to say G is a open localic group and I'm going to work through some of the consequences of an assumption that G of lock is equivalent to lock category of locales over some elementary topos E so E is some elementary topos category of locales over it what's going to happen if I do make the category of locales with a G accent G action the same thing as that category of locales the first thing to observe is that as I said in the opening remarks that we can embed E as the discrete objects relative to this category and I can do exactly the same thing over here because I have a sierpinski like object in here it's just sierpinski with the trivial action on it so I can define discrete and open and I can look at the discrete objects relative to this category it's a lemma but it's relatively easily to see by the technique that I've just described in terms of describing natural transformations relative to G's back in terms of natural transformations relative to lock by unwinding the theorem I just or proposition I just proved you can actually check that the discrete objects relative to here are the same thing as BG so just the usual thing a set with a G action on it so the objects are just discrete locales so with that information in play we know that these two things there's an equivalence here and because they're equivalent that's that well we know that's a topos anyway but that's a topos it's the same as that topos and therefore they're going to have the same categories of locales but we also know from the well-known Joellen Gini result that for any G there's actually a geometric morphism here which is an open suggestion that's why I assumed that G was open in the beginning that's an open suggestion and it's therefore of effective descent and it's well known that the statement that is being effective descent means that the locales over BG are given by G objects relative to the group that you get by pulling back these two localic geometric morphisms against each other so that pulls back in in the category of autopos is to B of G hat where G hat is the elemerdike as the etal completion of G so this actually is isomorphic to the sorry equivalent rather to the G objects relative to the sorry G objects on the etal completion of G now although I'm missing a little bit of detail you can also show that these these adjunctions are effectively over lock there's a technical aspect of that so using that adjunction in the case lock you actually these equivalences commute with lock and then you can go okay well if these are equivalent over lock they induce a so monad on lock and therefore G is actually isomorphic to G hat so it has to be etal complete so since there are examples of groups localic groups that are not etal complete there are examples of categories here's one if G is not locale complete that has this double exponential feature to it and that was the main result I wanted to get across and I don't know do I have a two or three minutes yeah so to make it a bit more real because given time constraints haven't really been able to make the connection as precise as like as I'd like but okay that's very interesting but that doesn't you know what's allowing you to call that thing a category of spaces there are axiomatic approaches to locales so you say I've got I've got an order enriched category it's got finite products finite co-products co-products are pullback stable I've got a Sipinski object which is an internal distributive lattice it classifies open and close I've got double exponentiation and I've got some other technical conditions around how this behaves interacting with equalizers and reflecting isomorphisms and things like that with those axioms in play you have a category which you can prove that proper maps are pullback stable open subjections are of effective descent the weekly closed subgroup theorem holds the Hoffman-Misloff theorem holds you can do topological lattice theory so there's an awful lot that you can do which I would call locale theory but we've now shown that you can do that locale theory without it actually being a a category of locale so that's why I've headlined it category of spaces that are not categories of locales any questions or comments not a real question but I'm reminded of combinations combinations in computer science okay what's that why you'd have to give me a bit more for the context there I mean it wasn't proper question anyway so the question would have been are there any relations or combinations ensemble I think I'd have to look up I mean just out of interest because it may be related the the double power the thing that I've expressed it really comes from the lower power locale lower power and upper power constructions and those were originally developed in domain theory in in theoretical computer science so it actually it actually come well the sort of the constructions here actually come from theoretical computer science but I'm not familiar with continuations per se can I make one more comment unless there's a another question the the other thing that's sort of even more interesting about it is that you can for each topos I've said that you can you've got your your locales e that works for each topos but you can also for each geometric morphism you can have an adjunction between locales and that adjunction commutes with the double power construction wherever it wherever I've written it down and that actually characterizes geometric morphisms so you can actually embed the category of topos is in this category of spaces and it's it's fully faithful and I guess what this is saying is that it's you know there's more of them than there are of geometric morphisms and I've done some preliminary work and some of the properties of geometric morphisms seem to work in this in this larger category yeah I don't think yeah there's a good question and and you know you could take the view oh well you got it wrong you know you haven't captured locales I I'm actually taking the opposite view I think it's a it's a more exciting and interesting thing and another thing is that one of the key results of topos theory is that when you've got a localic groupoid the points of the localic groupoid correspond to the principal bundles relative to the etal completion but in this case once you've embedded it in this larger large category you get the principal bundles without having to go to the etal completion so my intuition is that it's actually in some sense better yeah I guess my question is what characterizes locales among yeah there's as far as I'm aware there's not a um there's not an elementary characterization and um if you speak to Thomas Stracker I think he presents that as a fact you can't do an elementary characterization I've got some slightly awkward non-elementary characterizations but I've never been able to tie it down and I have been for a number of years so I guess this this sort of coming along last year or early early part this year has made me think okay well actually I'm quite happy to step outside locales and see what it looks like okay thank you