 OK, well, first of all, I'd like to thank the organizers for the invitation to take part in what I'm finding is a very stimulating conference. So I'm going to be talking about an additive categories. And I'm going to be talking about this picture here, which I will explain what's on it gradually. These are all additive categories of some sort or another. Yeah, so let me say what they are. So this is top diagram. These are three two categories. So the first one, this is a category of small or skeletally small abelian categories with exact functors, as the one arrow is. And natural transformations are always the two arrows. This is the category of the two categories of locally coherent growth and deek abelian categories. So those are growth and deek categories which are locally finitely presented. They've got a generating set of finitely presented objects. And those objects are actually coherent. And one definition in this context is that every finitely generated sub-object should be itself finitely presented. And the maps are, well, they're, I've called them coherent morphisms. They are just analogs of the adjoint pairs where the left adjoint is exact and preserves coherent objects. So it's really a, you've seen a picture very like this in the talks on Monday and Tuesday, where talking about general toposes and models of other, and the same situation, but for other kinds of logic, in particular, regular logic. It's the most relevant one here. OK, and then this third one, the objects are the so-called definable additive categories, which I will define. And the maps, they can be regarded either as model theoretic interpretations or simply as the functors which commute with direct products and directed co-elements. OK. And so these categories are basically equivalent or anti-equivalent. And this underneath is showing you how to get from one category to the other categories. And I will explain the various notations. The bit that's missing is a little too long. This is ab, that's missing. So this should be the exact functors from A to the category ab of abelian groups. Yeah, so I mean that the picture has been developed, essentially, within model theory of modules and representation theory itself, module theory itself. But it really is an analog of something that can be seen in the topos world. There are all sorts of points of parallel, broad and also giant level of technical details where you prove things. OK, and so part of the picture will be a special case of the regular logic picture. But there are some additional features in this case. And in particular, there's a duality that runs through the whole picture. OK, right. So I will begin with the original example of a definable category, which is just a category of modules over a fixed ring. So we fix a ring with one or just take any skeletally small pre-artificate category. So the ring is the case with that. There's just one object. So it's a category enriched in abelian groups. So you could just take, in fact, this notation means the category of finitely presented right R modules. OK, so mod R, this is a category of all right R modules. In other words, functors are always additive. This denotes a category of additive functors from R op to abelian groups. And little mod is for the category of finitely presented modules. Finitely generated, finitely related. Alternatively, a module is finitely presented, the general definition, if the covariant representable functor commutes with directed co-limits. OK, so yeah, so this is all coming from the model theory of modules, which started with van der Schmeyel's proof of decidability for abelian groups. And there were developments in the 70s. And then particularly, the power proved PP elimination of quantifiers, which I'll state. And that vastly simplified a lot of the existing proofs and allowed a lot more developments to take place in the context of the model theory of modules. And then, Siegler, in particular, produced a paper with a lot of new ideas and results at the end of the 70s. Well, he did the work about 1980. OK, and then the subject continued to develop, making more connections with representation theory. And particularly, with the kind of representation theory that was started by Auslander and Auslander Wrighton, where they used a functorial approach, a functor category approach, to understand modules over a ring. Yeah, so the idea there is that you want to understand, say, the finitely presented modules. So rather than investigating them directly, you take mod R to be the small, pre-added category. And you look at the category of mod R modules of functors from mod R to Abelian groups. And that turns out to simplify matters, though it looks as if it should complicate them. OK, and then, again, so there were more connections being made with the functorial approach to model theory. And then Evo Herzog realized that the duality, which had been seen in part, actually extended further into the picture. And he also introduced this category of PP imaginaries, which I will define. And that, again, changed the viewpoint a lot. OK, and so there's been a gradual process where this picture that I put up at the beginning has emerged. And it really only became clear not that long ago. Right, so a little bit about model theory. And this results PP elimination of quantifiers. OK, so in model theory, you choose kind of structure. And what we're doing here is we're fixing a ring. Or it could even be a small pre-added category and we're looking at modules at functors from that to Ab. And then we're investigating those structures. We set up an appropriate first order language. And it will be a finite ray language, OK, because we want to use compactness theorem here. So it's a finite ray first order language for our modules. You actually, I mean, there's various ways of setting that up. We don't really need them. We don't need to say explicitly what they are. Right, so in model theory, you're interested in the definable subsets of a structure. You're interested in the structures of modules. You want to know what the definable subsets are. So what do I mean by a definable subset? So we, well, solution sets of equations, OK? And then you, so basically, solution sets of equations, there's no relations in this language. So let me say this for languages without relations. Solution sets of equations, finite Boolean combinations, intersections, unions, complements of those, and projections of those. And those are definable subsets. I mean, those operations correspond to the logical, the Boolean connectives and or not, and the existential quantifier. So you're interested in what you get by looking at solution sets of equations and projecting them. And we're going to look in particular at the so-called PP for positive primitive, but they are actually just regular in the earlier terminology we saw earlier in the week, formulas. They're formulas of this kind. So the context, the three variables, are the tuple x bar, supposed to be x1 up to x, and a tuple of variables. So you just look at a homogeneous linear system of linear equations, our linear equations, where r is the ring, and you project that. You project some of the coordinates. OK, so projected systems of homogeneous linear equations, so the solution sets are always going to be subgroups of your module, or powers of the module. And the basic result in the model theory of modules that proved by Bayer is that every definable subset of a module, every subset you can get by not just using this kind of formula, but also taking complements, unions, is actually a finite Boolean combination of these simple, these PP-definable subsets. And we should allow ourselves inhomogeneous systems to allow parameters from a module to appear, so that it's actually cosets of such PP-definable subgroups that are basic definable sets. OK, and the result says more, as I've indicated, but this is the part we just need. There's a natural ordering in PP formulas. So notice that one of these formulas, it defines a functor, an additive functor from the category of our modules to the category of the Boolean groups. OK, so you take a formula phi, and you define a corresponding functor from the category of modules to add, just taking a module to the solution set in that module of that formula. And because the formula is PP, the solution sets are preserved by morphisms, so it is a functor. So we order our formulas according to the order of the solution sets or the functors. So essentially, we write psi is less than or equal to phi. If the, in every module, the solution sets are ordered like that, in other words, psi is a sub-functor of phi. Right, so the corollary is that the model theory fits well with the algebra. So what I mean by that is if you're doing model theory, the category that you naturally work in has the objects that you're interested in as the objects. But then the maps tend to be the ones that preserve and reflect the definable sets, in other words, the elementary embeddings. And there are very few of those in general. But because here the PP formulas are due to define, well, they're preserved on the solution sets preserved by morphisms, it does mean that we can stay, for most of the time, in the ordinary algebraic category with just the normal category of modules. So at first sight, it seems if you're going to do the model theory of modules, you have to work in the category of modules with elementary embeddings. But in fact, most of the time you can just work in the ordinary category of modules. If you do want your maps to reflect solution sets of PP formulas, then you need the pure embeddings. You should work in the category with pure embeddings, where you say an embedding of modules is pure. If for every PP formula, phi, you have the solution set in A, the first module is just the solution set in the larger one intersected with A. Is it the same as universally injective, which is sometimes? Universally injective. Which means when you tensor it with another module on the left? Yeah. So it's equivalent to that if these are right modules, this is equivalent to saying that if you tensor with any left module, you get an embedding of a B-link groups. Yeah, it's equivalent. Yeah, yeah. OK. And yeah, so the model theory changes become simpler in a sense and more closer to the algebra, because even the elementary embeddings, you can replace them by the pure embeddings. And in model theory, you tend to look at so-called saturated modules. And here, it's enough to look at the pure injective modules. And I'll talk quite a lot about pure injective modules. So the pure injective modules are there kind of, if you like, saturated for sets of PP formulas. If you've got a bunch of cosets, if you say a module is pure injective, well, two definitions. It's either pure injective meaning it's injective over these maps, or you look at, take any filter, any, say, filter of cosets of PP-definable subgroups. So such that the intersection of finite to the many then is non-empty, then the intersection of the whole lot should be non-empty. So it's a kind of completeness property. Right, so it turns out that these pure injective modules are really rather important in this context. Yeah, examples of pure injective modules. Well, OK, so suppose we're working with a billion group stick, the ring to be Z. So the indie-composable pure injectives, you've got the finite indie-composable modules. Let's look at the indie-composable. So integers modulo p to the n, they are pure injective. I'll write them down. So over Z, the indie-composable pure injectives, you've got the integers mod p to the n. You've got the proof of groups. You've got the periodic integers. And you've got the rationals for various p and n, OK? And in general, any module which is finite dimensional, which has an underlying vector space structure, and if it's such it's finite dimensional, then it's pure injective. Joules of modules are pure injective. Homejoules of modules are pure injective. OK, so they're quite common. Right, so we've already heard about imaginaries. I'll say what they look like in this context or how we want to adapt them in this context. Right, so imaginaries were introduced in model theory in the 70s. And they were regarded as a bit strange to begin with, but people gradually began to accept them and use them. So for example, you start with a module, which is once you're using a one-sorted language, all your elements say they're in the same place, the same kind. But it's useful to add not just integrals of elements, which you can regard as a new sort, but other things. You can take elements, you can quotient these by definable equivalence relations and add those powers of the home sort factored by definable equivalence relations with new sorts, new kinds of elements. So that's become very common within model theory to work not just with the original structure, but with all these imaginary sorts around about it as well. And the model theory of modules became very natural not to add just a few of them, the ones you need, but actually to look at the whole category of all these sorts, just add them all at once. Because it turns out to be a nice category. OK, so what are they? So because we are working in this additive context, we want this additive structure to be preserved but in our new sorts. So that does force us to use only to add new sorts made from PP formulas. So essentially, if you have a PP formula, you take that set that it cuts out to be a new sort. You can factor it by definable equivalence relation, which again should be PP defined. In other words, it's equivalent to factoring by subgroup. Remember these five-side defined subgroup solvice? So essentially, you just add some new quotient subgroups. The module is there. That's the basic sort, but you have all these extra sorts around about. OK. And though it turns out to be extremely useful to you to have those sorts there, and the category of them is nice. So examples of sorts that you get, you could take, yeah. So if A is a finitely presented module, then HOM, a blank, is a new sort. Next, if A is FP2, meaning it's got a projected presentation where one, two, three terms are finitely generated projectives, then X1, A blank, also is a new sort. And so for example, OK, so a lot of homological. So it's saying that if you now put a module in there, you've got the module as a basic sort, but attached to it, you have HOMs for many finitely presented modules. These are all new sorts. The Xs, a lot of X groups and so on. So they're all somehow there implicit in the model theory of the original module. Can you say what the formula would be for X1? Yes. But I mean not maybe very quickly. But yeah, you just write down the projected presentation and you can write down the formula explicitly from that. Yeah, yeah, yeah. OK. Yeah, so let's look at the category of these PP sorts, these new imaginaries. So the objects are just PP pairs. So take two PP formulas, phi containing psi. That essentially we're thinking of it as a quotient, where we mean the quotient is what we're adding as a new sort. And the maps from one such category sort, one such object to another, are given by the relations which define, which are actually functional, the relations from the first sort to the second which are functional. OK, so again, you saw those on Monday or Tuesday in the other context, according to various contexts in the topos context. OK, and the notation we'll use is, OK, indicating the ring. This is a category of PP imaginaries. And the EQ is the last notation for adding the imaginaries and the plus indicates we're keeping the additive structure, just using PP formulas. OK. Right, so it turns out that this category is equivalent to the category of finitely presented functors on finitely presented modules, OK? So in other, so, I mean, mod R with a big M is a locally finitely presented category. So you're taking the finitely presented objects and then you're looking at the pre-sheaf category. This is really just a pre-sheaf category, right? Except you've got AB instead of SET, because we're in the additive world. So you're taking the finitely presented pre-sheafs on the finitely presented objects, OK? And so this is, I think, something that's probably there already in the, I mean, there are certain versions of this in the SET-based world. So you, so the imaginaries have to do with the final equivalence relation, which are provably equivalence relations, absolutely. I'm not, just, which are, the provable doesn't matter. It's just, just definable equivalence relation, just formulas which define equivalence relation. Well, which, which approve, yeah, which, which modulo the theory of modules approve, yeah, define equivalence relation, yeah, yeah. Yeah, we're working all, that's right, yeah, we're working all the time relative to the theory of modules over that ring. So the, the fore, when I say that something defines an equivalence relation or that our relation is functional, I mean, what, you know, it's provable within the, or it's true within every module that it does that. Equivalently, it's provable from the theory of modules that it does that, yeah. OK, and yeah, this, it turns out this category is also the, the free abelian category, which a notion of fried, it's free abelian category on R, so you can start with a ring or any small pre, small, certainly small, pre-additive category and define free abelian category. You mean the category of one object and an R, not the module of R, or R. It's not the modules over R, yeah, it's, it's, yeah, that's right, I mean, if you start with just a ring, then it's free in the sense that, so it's one object and the ring is there in the movement. Yeah, yeah, so I'm thinking of a ring as just being a one object, pre-additive category with the elements as being the endomorphisms. So the free abelian category of R is an abelian, a morphism to an abelian cat, small abelian category with the property that if you have any morphism to here, to an abelian category, then there's essentially a unique exact function there filling in that diagram. Yeah, I mean, Freud proved it more generally for small pre-additive categories with an exact structure on them, and this then would be preserving the exact, taking the exact structure there to the normal exact structure on A. Okay, so yeah, so what we've got then for at least, for starting with the beginning case, starting with the category of modules as our definable category, we've got this associated, skeletally small abelian category here, and then here we've got the category of all the functions from mod R to ab, which is just the end completion of that thing there. Okay, so that's that initial picture I put up, that's one example then, where I stick in that category at that point. And we get the general case by localizing. Bear in mind that R doesn't have to be a ring with one object to kind of many objects. So basically we just localize it at torsion, well, let's say our subcategory is there, torsion theory is a finite type there, and I'll say what it looks like here when we do that. But before doing that, I want to talk about a couple of associated structures that we have coming from the model theory. First one is the seagull spectrum. Right, so it's a topological space. You look at the set, it is a set of isomorphism classes of indecomposable pure injective objects. Okay, so in the case of the integers, you'd be taking the, if the ring was z, you'd be taking these, these, these and that. Okay, and you'd apologize that set. You take for a basis of open sets, they are given by the PP pairs. So you take all the points in that space where that pair is open, where that sort is not just the zero group. Okay, and that's going to be a basic open set, and Seagull showed this gives you a topology and it's a compact, or some would say quasi-compact space. It's certainly nothing like Hausdorff. So it's a compact space, these are actually the compact opens. Yeah, I should say if ours are ringing with one or more than many objects, it's compact, otherwise, if you've got to inflate many objects in R, it won't be compact as a whole, but these will still be, the basic open sets still will be compact. And this turned out to be extremely useful space for the model theory of modules in various ways. Okay, I should also say just a couple of comments about that space. I mean, there's a lot about it, but just a couple of things. So one is, I've been talking about right modules. I'm not assuming it's all, the ring is commutative, so we've got a category of left modules, which could actually be quite different in its kind of finiteness conditions and so on. But there is a reasonable duality here, so you can look at the, this was at Seagull's spectrum for right modules, is at Seagull's spectrum for left modules, and they just turn out, the hertz are proved that at least as locales, if you just look at the open sets, you get a nice morphism between those. And in fact, I mean, it may be that even at the level of points, that's a lot of true as well. Maybe it's literally a homomorphism from right to left, but that's only been proved under some conditions on the ring. There's also another topology on the space, which is the Hoxter dual. So that construction, which Hoxter introduced for spectral spaces, which these are not, but which you can still make the construction. So you take the basic open sets, sorry, you take the compact open sets, you take the complements of them, and you declare those to be open, and you're new topology. So you do that, and you get to the Tegeler topology, and you get a new topology, which is much more like the risky spectrum of a, say, commutative Natherian coherent ring. And in fact, it is the, if you take the definition of the spectrum of a commutative Natherian ring, it may say it in terms of the category of modules, sort of placing primes by indigmosable injectives, et cetera. Then, let's say that's the Gabriel spectrum. Then it is the Gabriel spectrum of the functor category. Although note the, yeah, I mean, that's on the other side there. Quotonica billion, you mean AB5? Sorry? Quotonica billion, you mean satisfying the axon AB5? Yes, yeah, yeah, yeah, yeah, yeah. Yeah, so I mean, this category here, it is just a module category. You really should think of it as a module category. So functor. The category is the category of R modules, the category of functors, the Gabriel spectrum of the category of functors from R modules to R, or? Yeah, sorry, say it again. You take the Gabriel spectrum, I don't remember our definition. Yeah. The category of R modules, or the category of functors. The category of functors. Yes, you take the category of functors from phonetically presented left R modules to AB. You look at, so the points of the space of indecomposable injectives, isomorphism types of them, and the topology is given by, you take the phonetically presented objects and that phonetically presented functors there, and you take the harms from those, you know, one of those blank, to give, I forget if it's the open or closed set. You take those to give you the sets of the topology, the basic closed or open sets of the topology. Yeah, I mean, if you apply this construction, if you start, in fact, with just R there to AB, so talking about the category of modules over a commutative Notherian ring, then this exactly gives you the normal prime, the prime spectrum with the Zyrtysky topology. If the ring is commutative. Sorry? If the ring is commutative. If the ring is commutative, yeah, that's right, that's right. The category of functors from R modules to AB, phonetically presented, so you mean additive functors? Yeah, yeah, I also mean additive functors when I talk about functors. But they sound exactly like this. No, yeah, that's right. They don't have to be exact, yeah, yeah, yeah. Just additive functors from R module to AB, such as the take a PP formula and evaluate it. I mean, they don't have to be, that doesn't have to be exact. So what is the magic at the level of points? I mean, because Gabriel's spectrum, you take of the functor category and then you have pure injectives is the same as pure injectives at the beginning. Yeah, that's right. So you have this, I mean, the reason this works is an embedding of the module, the left module category, so mod R embeds into the category of functors on, it's the tensor embedding, well, the tensor that was mentioned earlier, embeds in that functor category, but you just take a module and you take it to the functor and tensor blank. Okay. So that's the embedding that makes this work. You embed left R modules into functors on finally presented right R modules. And that takes, yeah, so a pure sequence here is pure exact if and only if its image is exact, as I said earlier, and the pure injectives here exactly correspond to the injectives there. So that's the point, yeah. So the pure injectives here, that's the points of the Tegla spectrum give you exactly the injectives, that's the points of the Gabrielle spectrum. So you get more, like when you take a commutative material ring for simplicity, then if you take the Gabrielle spectrum of R modules, you get a spectrum of R. Yes. And when you take the Gabrielle spectrum of this stuff, you'll get a lot of points. Within it, you will have the closed set of the original injectives, but you get a lot more points, yeah. If you take R commutative, this, yeah, I mean, the injective R modules are pure injectives, so there are points here, but in general, there are a lot more points, yeah. So we've lifted the definition of the Gabrielle spectrum up one representation level. Okay, yeah, so the other structure is this, well, looking at the ordering and PP formulas, they do form a lattice under an intersection and sum. So it's interesting to look at that lattice. And you see the duality between right and left happening there as well. Yeah, so, yeah, so all these PP formulas are ordered by inclusion of their solution sets, or if you like, they're just, I mean, every PP formula gives you a sub-functor of the forgetful functor, or if it's an N-free variable, sub-functor of the Nth power of the forgetful functor from mod R to AB. So it's really just the lattice of finally generated sub-functors of the forgetful functor from mod R to AB. I mean, this is a, that's a coherent object, so they're actually finally presented functors. It's also equivalent to that as N-pointed, finally presented modules, so R, N blank, the slice category from RN to mod R, to little mod R. Okay, so there's a duality on formulas, it's explicit. If you've got a PP formula, you can, if you're thinking of it for right modules, you can write down a PP formula, it's dual, apply it to left modules, and that gives you an anti-isomorphism of the lattice for right functors and lattice for left functors. And you do it twice and you get back where you started up to equivalents. Okay, so this was a, yeah, I mean, one, I guess the place where the duality in all this was noticed, first of all, but it actually extends through everything. And it's obvious when you get to the final point, because that two category of smaller billion categories has an obvious, an obvious automorphism of order two, namely, just take a, replace each category by its opposite. But it's less obvious elsewhere. So the duality is saying that the three are billion category generated by a ring. Yeah. And the opposite of the three are billion category generated by the opposite ring. Yes, yes, yeah, yeah, the opt, yeah, yeah, yeah, yeah, yeah. Right, yeah, so this is talking about what happens, okay, when you localize these functor categories, but what happens to the definable category on that side of the diagram, the triangle? So suppose we say a definable subcategory of mod R, it's given by adding some extra conditions, which say that one PP formula, take a PP payer. If you add the condition that says that phi also is less than or equal to psi. So you insist that this sort be zero. These were mentioned again earlier in the week, they're just adding sequence. So I guess with the notation from then, you'd be doing something like adding a condition to get, to say that here is closed, you would add that, I guess, to close it. That sequence needs a regular formulas. So you just add these extra conditions and you look at the modules which satisfy those and that's a defined, they form the full subcategory and those is a definable subcategory of the category of R modules. And there's an algebraic characterization. Categories close under direct products, direct co-limits and pure submodules. And these are really the right contexts for doing model theory, additive model theory, if you want the nicest results anyway. And things to work well. So it turns out, so I mean, in that picture, do I have it? Yeah, so imagine the whole picture. So this is the final category. Yeah. It turns out the definable categories correspond to the close subsets of the TECO spectrum. But I was just sort of obvious now once I've said all this. Okay, the close subsets are just given by saying certain pairs are closed. So the definable subcategories of mod R correspond to the close subsets of the TECO spectrum. Yeah. Okay, and so now this is what happens when you localize. Originally I had mod R, the category of all R modules there. Suppose now I take a definable subcategory by adding certain sequence, certain axioms. Then the effect on the category here is to localize it to factor by the set subcategory consisting of all the functors, the finitely presented functors, which are zero on every object of your definable subcategory D. So you look at the, you can fix the definable subcategory, you look at all the functors F. Well, now F is strictly speaking defined on finitely presented R modules, but every module is a directed co-limit of these. So there's a unique limit extension of F to all modules. So you ask whether F is zero on every object of D. If it is, you put it in the set subcategory. And that's equivalent, then this is a locally co- Yeah, these categories here are locally coherent without forget the torsion theory. Mod R-AB is a locally coherent category, in fact. It's nicer than a general ring category of modules. So the, and you get the torsion theory, set subcategory here gives you finite type torsion theory there. So the picture localizes. And this gives you the general case, in fact, as long as you start, allow yourself to start with a ring with many objects. So you get all, you get all small of being categories up here, for example. Yeah, and the, you can also think of the imaginary's interpretation, sorry, of this category, this was a category of PP imaginaries. That also localizes in the sense that you just, you have the same objects. And as was discussed before, you've got the same objects, but now you have more conditions, more sequence. So some relations, which before, just defined relations, not functions, now may define functions. So you get more functions. So that's your localization, your category of imaginaries for the definable subcategory D. Okay, and yeah, as I said, every small abelian category arises in this way. Right, so that's general picture again. Okay. Yeah, so, I mean, briefly, how do you get from one to the other? If you start with a small abelian category A, you, the corresponding definable subcategory are actually the exact functions from A to AB. And then from A to here, it's just the lim, the, well, you can think of it as a flat, right, A modules, or it's just ind, ind of A is another way of writing it, just the intercompletion. And yeah, well, you could, AB's here stands for the absolutely pure objects, otherwise known as FB injective, the ones with X, one with finitely presented objects being zero. Okay, yeah, so just a couple of things about that picture to point out about it. One is that a definable category D can be recovered from the category of imaginaries. So this is the, this would be the, I guess you take the syntactic category, so we were talking about regular logic here, so you'd be taking the syntactic category and then the effectivization, but that's the same thing in the earlier language. Okay, so yeah, you can get the definable category D from this category of imaginaries. And in other direction, if you have, this was less clear, if you have a definable category D, you can actually recover the, and you're not, you're just given it purely as a category, not, you're not giving, if you're given it as a representation, the definable subcategory, you can get the definable category. It's category of imaginaries by localizing, but if you're just given it as a category, how would you, with no particular representation of it, how would you get the category of a imaginary, so it turns out to be just the functions from it to AB, which commutes direct products and direct limits. And these are, this again was less, not obvious that these are just, well not obvious, but maybe in the general context it's more clear. These are just the interpretation functions in the model theoretic sense. So more generally, if you have two definable categories, C and D, then the model theoretic interpretation functions from one to the other are just the functions which commute with direct products and direct limits, directed coordinates. And yeah, they are in natural two-categorical bijection with the exact functions from the corresponding marginally, syntactic categories, or if you want, if they're functor categories. I'm curious, why do you need the definable, it seems that everything there is true without assuming that D is the fun? So what else would, just, additive? You say as the funnable category D can be recovered, but what about saying, should the additive, small additive category can be recovered? I don't know, well, I mean, yeah, I think you need something to get that, but I'm not sure. Well, because, I mean, well, I mean, we know that any functor of that, if you have a small abelian category here, then the category of exact functors from it to AB will be a definable category. So if you want exactly this construction, you're gonna get the definable categories. I mean, these do include locally, finitely presented additive categories, includes finitely accessible, more generally finitely accessible categories with products, all of those are definable categories. But it's also got, there are also categories there, which are not finitely accessible, included under the definable categories. Okay, so yeah, I mean, this is the general framework. I want to say a little bit about just how it's used in practice. So on the way, a lot of the development happened within the model theory of modules, but a lot of the applications have been to, well, two modules, and in particular to representations of modules over finite dimensional algebras. So I'm gonna, oh, before that, yeah, just point out duality runs through the whole picture. All right, I'll just flip back to the picture for a moment. There, you have an obvious duality there. Take any Abelian category to its opposite. But that transfers into the whole picture by these equivalences, through these equivalences. So for every definable category, there's a dual definable category, which for right modules is left modules. And yeah, and for the locally coherent categories, that was a, what can we call, I think the conjugate category. Yeah, and on the Abelian, small Abelian category side, if you think of them as functor categories, this was already used a lot by Auslander and Reiton, this equivalence of functor categories. And Grusso and Jensen also developed it in the work, Jen, Auslander and Reiton used this a lot for finite dimensional algebras. Grusso and Jensen developed some of the general theory around there. Okay, and the duality does seem to do a lot that, I'll give you a lot that maybe you don't have in the general case. Okay, yeah, so using this in some regular contexts, so let's take the representations of finite dimensional algebras, so by which I mean just an algebra over a field, which as a vector space is finite dimensional. And in fact, all the comments will work for art and algebras, which are algebras whose centre is an Artinian ring which are finitely generated as modules over the centre. Okay, so what about pure injectives? The indie composables in particular are the points of the spectrum. So all the finite dimensional modules. And you mean over a field? Over a field, over a field, yeah, yeah, yeah. Cuted field, just to, yeah. Okay, so yeah, so finite dimensional modules are pure injective. If the algebras are finite representation type, meaning there's only finite for many, then that's it. But otherwise there are some, I remember the space is compact. Well, I need one more fact. But yeah, I need to say now, these points are actually isolated. They're open points of the tequila spectrum, the finite dimensional points in this case. So there will be some more points by compactness if there are infinitely many. There will be infinite dimensional pure injectives. Okay, so this is an obvious thing to take a finite dimensional algebra or a class of them and say what is a tequila spectrum? I assume somebody's already figured out what the finite dimensional representations are. What about the limits of those? Because essentially the infinite dimensional pure injectives will be limits of those in the tequila spectrum. Closures of those, or sets of those. Okay, another problem within this area is to determine how difficult it is to classify the modules over the ring. So you have these notions of tame representation type and wild representation type, which and various refinements of that. The idea being that if you have an algebra of tame representation type, you can probably classify the modules in some reasonable way. And if it's wild, well, if it's wild, there are themes saying that the classification problem for that ring contains the classification problem for lots and lots of other rings, essentially any other ring. So looking at how, looking at measures of complexity of a category of modules is another problem that people have looked at here. Okay, right, so let me first give you some measures of complexity. So the first is the M dimension for the lattice of PP formulas. Okay, so we'll just take the PP formulas in one free variable, in other words, take forgetful functor from more finitely presented modules to a billion groups and take its finitely generated sub-functors, they form a modular, they form a lattice, sub-lattice. Okay, so there's a general procedure for giving dimensions on modular lattices, a dimension on modular lattices. So you take your lattice, you look at the interval switch or a finite length and you collapse them. Okay, you look at the quotient lattice, that will again be a modular lattice. You repeat the process, trans-finite play. Okay, so that gives a note and eventually the thing maybe collapses to a single point. So that gives you a notion of dimension for a modular lattice and we're going to call that M dimension and we're going to apply, well, M dimension when applied to the lattice of PP formulas. It's a bit like a cruel dimension in the sense of Gabrielle and Wrenchler and people who developed that, but it's got no direction, unlike that dimension. Right, another measure of complexity, we're just fine with the functor category. So we take this finally presented functor category and again, we say we're going to collapse, we're going to take the set of category of functors of finite length, factor them out. You get a new abelian category. Take the objects of finite length there, factor them out, keep going, trans-finitely. At some point you may end up with no, as a category with no functors of finite length. So you say the dimension is infinite, undefined. Otherwise it stops at some point and that's, you count how many steps you had to take and that's the dimension, the KG dimension. I should have said as well, with the lattices, you might have got to a point where the quotient lattice had got no intervals of finite length. In other words, it contains a copy of the rationals as an ordered set. Then you say the dimension is undefined. Okay, I'm saying these very quickly, but just to give you an idea. Okay, and then the third dimension is cantibendix and rank, maybe more familiar of a topological space. So here you've got a topological space. You look at the isolated points, the points that are open. You throw them away, you've got a closed set that's left. You look at the isolated points there, you throw them away and you keep going. And if your original space was compact, you will actually stop at, well, okay. So either you reach a point where there's no isolated points, then the dimension is infinite, or you reach a point where you've got rid of everything and then you give the space and also points, ranks according to when they were thrown away. Right, so these dimensions, I mean, they sound rather similar in some, especially the first two and that's because they are the same. So the Krull-Gabriel dimension of the functor category is the M dimension of the PP lattice. And conjecturally, well, it could be that it's also equal to the counter-Bendixen rank of that teacup spectrum. And that's certainly true if these dimensions are defined and are also, in fact, in lots of other cases, but it's still open in general. So this seems to be essentially one dimension. So I just want to compare, say, give some results comparing that dimension with the complexity of the representation theory of a finite dimensional algebra. Okay, so a bunch of results. Well, I should say about, sorry, the numbering here is maybe looks like, I can't do alphabetical order very well. There's a bibliography. The number means that you find the reference on the page with the reference and then you count down. I was unsure how to put along bibliography and more than across more than one frame. So I've split into different bibliographies, which is why the numbering's a bit strange. All right, okay, so take a finite dimensional algebra. And yeah, I've said already, the finite dimensional indie-composable modules are exactly the isolated points of the spectrum. And in fact, they're dense. Together, they're dense in the space. So every indie-composable pure injective is in the closure, the definable subcategory generated by finite dimensional points. In fact, a stronger result is that it's a direct summand of a direct product of finite dimensional modules. Okay, right, and then the, what about the value of the Krull-Gabriel dimension of the ring? So I guess Geigler, actually, I should have listed him somewhere here. He was the first person to look at this dimension for finite dimensional algebras. Okay, right, so what can we say, what's known so far? So Krull-Gabriel dimension zero is equivalent to the ring being a finite representation type. Dimension one turns out to be impossible for outer algebras. It's certainly possible for other rings, but not for outer algebras. Dimension two occurs, so tame, red-ray algebras. So path algebras of extended dink and quivers are, they have Krull-Gabriel dimension two. And various, some other classes of algebras. It was a conjecture for some time whether the only possible values in this context were zero, well, one, two, and infinity. In fact, there are finite, any finite value apart from one can occur. So, yeah, Jan Schreuer produced some examples. And then, in fact, we have got some more complete results there. So any finite value apart from one can occur from the Krull-Gabriel dimension of these algebras. And that's a dimension that is preserved by these interpretation functions between the final categories. So you can say, if you've got an interpretation function, one of these functions that commutes with direct products and direct limits between two module categories or subcategories of the final subcategories of those, then you know the dimension of this one's at least, this is a nice functor, at least, if it's essentially zero kernel. You know the dimension of this one's at least as large as that. Okay, and this conjecture about the value might always be finite for these algebras. It's known to be undefined for wild algebras and also for some tame algebras. So there's a conjecture that perhaps finite Krull-Gabriel dimension corresponds to domestic representation type. Okay, so I mean this is a lot of information, but just to give some idea of what some of this looks like in a particular context. Okay, so yeah, just to say that notion of something about representation embeddings, how you compare the complexity of one module category with another. If you take two finite dimensional algebras, so a representation embedding from the finite dimensional modules over one to the most of the other. So it's a functor which is exact, takes exact sequences to exact sequences, preserves and decomposability, and reflects isomorphism. So if two things are, the image of two things over here are isomorphic, they were already isomorphic to begin with. Okay, and such a functor will have, we'll actually be testing with a suitable bi-module, which makes it a bit more explicit. Okay, and so yeah, and the idea of representation embedding is that the representation theory of R is at least as complex as the representation theory of S, because this is essentially taking mod S and embedding it into mod R with no loss of complexity. Those conditions can be seen as saying that. Okay, so yeah, when we'd like to know that a representation embedding preserves these dimensions that I've mentioned, and that and also some others, and that is the case, particular cases were known before, but yeah, in general, in general representation embedding will induce an embedding of lattices, and hence the dimension of, they say if you have a representation embedding from mod S to mod R, the Krull-Gambriel dimension of R will be at least that of mod S. Okay, so I think the rest is bibliography, yeah various pages of it, yeah. Right, so yeah, so I wanted to give you the general picture, but then also to say something about how these ideas appear in a specific context. So yeah, that's what I wanted to say. Have you a question? I'm curious about the terminology, why imaginaries? Oh, well, this is, well, I mean, I guess it started with Shilah, I don't know if Shilah exactly was responsible, but they were called the imaginaries very early on, because you're starting with a module, say, and now you're adding these new kinds of elements. So it's a bit like, yeah, constructing the imaginaries as pairs. They are not modules, so they are not concrete. They're, well, yeah, I mean, they're kind of, you thought you knew what a module was, it's just a one kind of element in it, but now you find all these other kinds of elements that can be, no. So I mean, they're elements, but regarding the module in a category with more sorts, yeah. So I think it was an analogy with the machine, yeah. Have you made an analogous triangle in the non-analytic setting? I haven't tried to write, yeah, I, yeah, you should probably do that. Yeah, I don't know the non-analytic setting as well. I was wondering in particular, do you think that the place of smaller billion categories is played by effective regular or rather pre-toposis? Yeah, I mean, it looked, so I did have a student, Philip Bridge, who looked at trying to move the additive stuff to the non-analytic, and yeah, he was looking at pre-toposis and so on, but he was finding the results we wanted, like these dimensions, he was trying to define the dimensions in the non-analytic case, like Krull-Garriel dimension or M-dimension, but he was really finding the lack of duality was a problem. And yeah, so he got somewhere with that. He got further with other things, so it seemed the lack of the duality was really making a big difference in the additive case. So yeah, I don't know. I've not thought about it since then. Just a small debate about Pp1S into PpNS doesn't seem correct. No, in fact it is, yeah, you do have, yeah, because what happens is that a representation in Bernie, you're testing with this bi-module, which is projective on one side. So you do have, it's essentially, this is in Bernie, it's a marita equivalence followed by restriction of scalars. And the marita equivalence is to n by n matrices over the original, so that's where the n comes from, yeah.