 Hi, everyone. My name's Thomas Payne. I'm here to talk about game co-monads and an equi-ranked variable homomorphism preservation conjecture, which is a bit of a mouthful, sorry about that, but the title just pretty much describes the two parts of the talk. So part one is on game co-monads and this is building on work. So there's one paper on the pebblin co-monad by Abramsky, Dawa and Wang, and then there's another paper on similar co-monads I think coming out soon by Abramsky and Shah. So those are the things that I'm going to talk about in the first section, just to sort of be able to discuss the application to a homomorphism preservation theorem, which is going to be a slight refinement of what's called the equi-ranked homomorphism preservation theorem, which was proved by Rossman. So just to set the scene, we'll be working in a category of relational structures. So we're going to have a finite vocabulary of solely consisting of relations and the associated category of sets which interpret those relations and morphisms between those sets. And then sometimes we like to talk about the category of those structures with an associated distinguished tuple and morphisms between those which preserve not only the relations but also the distinguished tuple. And we're going to consider the set of first-order sentences over the vocabulary sigma up to just first-order equivalents. And we're just going to consider that to be a poset. So we're going to identify things which are the same and have this order if one implies the other. And we would like to pick out at the beginning the positive existential sentences, which one's built using only exists and all. And also we'll talk about formulas with sorry, so we'll talk about sentences most of the time and then sometimes talk about formulas with some number of free variables. And when these formulas are interpreted by structures in C sigma with the tuples interpreting the free variables. So a nice picture we can draw between existential positive sentences and finite structures. Sorry, this category is a category of finite structures on the previous slide but I didn't say it. Is that of the canonical query and the canonical query is a first-order positive existential formula and it just picks out all of the relations that are true of a given structure. So we just take one quantifier for each element of the structure and then we just take the disjunction of all of the relations that they satisfy. And this is kind of you could describe this in a more abstract way. So it's the formula with these properties that is the largest, so it's the maximal positive existential formula according to this order that A satisfies. And we can phrase almost all there is to know about it by saying this is a surjective and full functor. So this is actually the sort of collapsing if you like of the category fc sigma. And this is an example so if you've got a functor, if you've got a morphism, then functoriality is telling you precisely that the canonical query of B should entail the canonical query of A. So what we're interested in doing is seeing how this picture works when we consider different fragments of first-order logic. So we can break down, first of all, by quantifier rank. So that's the nesting depth of quantifiers in a given sentence. And we can also break down by the number of variables as well as quantifier rank. So number of variables, you do have to do this funny thing where you quantify over a variable and then you at some point later introduce a new quantifier using the same variable. So that means you have to forget what it was before. So you kind of limited quantifier rank and then you have this other sort of limited memory or resource if you like. And we're going to see how this picture works that we drew before with regards to these smaller fragments. So the first thing we can do is define a sort of analog of the canonical query. And so it's going to have this property, it's going to be the maximal formula in ln plus that A satisfies. So that's property one and two. And we can define it in a much more direct way than before because ln plus is actually a finite poset unlike l plus. So we can just take the disjunction of all of the formulas that A satisfies. And this also works for infinite structures, which is nice. But this is no longer a full functor. It's still subjective. But sorry, I just sort of repeated it for emphasis on this slide. So we define this sort of 2n relation, which is kind of just the pre-image if you like of this relation here. And so a homomorphism is not implied. So just because the n canonical query of B implies the n canonical query of A, it doesn't mean there's a homomorphism from A to B unlike in the previous case with finite structures. So this is a kind of important relation in general in finite model theory and model theory. And so we're looking for the compositional structure behind this, if you like, in order to have a similar story to what we had before. So we can look at those structures whose pre-image, sorry, who mapped into ln plus under the canonical query. That's another good place to look. And those are structures with forest covers. So forest cover is just a forest on the same set. So forest cover of structure A is a forest on the same set. And you have to pick roots of the forest. And then you say that its depth is the length from the longest length of a path from the root to a leaf. And then if two elements of A ever occur together in a relation, then they must be on the same branch of the forest. So what's that got to do with restricted quantifier rank? Well, you imagine if you wanted to write, if you wanted to list the canonical query out in the same way that we did before, but this just tells you how to structure the canonical query. So you imagine just kind of writing it on this forest cover, and this condition that two elements of A who occur together in the same tuple must lie on the same branch tells you that you're not going to miss any relations. So this is quite important. And so this is called a structure with who, the least answer to that structure has a forest cover of depth N is called its tree depth. And this is an important parameter. And here's just an example. So I have my structure A, and I can give it a two forest cover. So if we look at, you know, let's say D is on the same, well D is on the same branch as everything. So B is on the same branches, C and A, which is good. And B and F, for example, are on different branches, but there's no there's no relation between B and F. So the relations are only between the things with arrows between them in the structure. We can put these together into a category. So we need to, you know, you could just take the subcategory of C sigma, which is structures with forest covers, but we get slightly more juice out of it than that by taking the category of pairs of structures and a forest cover. And then if you want amorphous in between them, you need amorphous and obviously of the underlying structures. And then you need to preserve what is the order that is induced by the forest cover. So really what you need is the branch relation being preserved. And we have then a pretty obvious functor that just forgets the relation. And what we're really interested in is the co monad that this will define on CN because this UN has a right adjoint and the co monad has a lot of nice properties that result in co monad. So in particular, so if we define EN to be this co monad, its co clisely category is the thing that kind of completes the picture, giving us this full and subjective functor using the N canonical query. So in other words, we if I had this relation, this relation is now captured by amorphism from ENA to be and co algebras again, because it is the because it is the co monad arising out of this particular adjunction, a co algebra will just be an end forest cover. So this gives us the compositional relation that we wanted and to boot it gives us a categorical description of tree depth. So in order to prove that such an adjunction exists, it suffices just to, you know, give a construction of it. And this is, you know, a lot of stuff to take in, but this is just a concrete definition. So you take lists of non empty sequences of length at most N, and you define the relations on us to be the set, and then you define the relations on it to be, well, you take, you say that two sequences are comparable, if one is a sequence of the other. So you can imagine this being like two elements lying on the same branch, if you like. And then you say that, but you also need to say that the last element, so this epsilon picks out the last element of a sequence, and the the relation should hold between the last elements. So there's kind of some intuition there that these pods maybe like sub trees of sub forests, perhaps of EN, are going to somehow correspond to co-algebras, why did they do correspond exactly to co-algebras, like we want them to. And what co-algebras and forest covers like we want them to. So sub trees are going to correspond to co-algebras, and they also correspond to forest covers, which is where that correspondence is going to come from. So this in terms of sets, it just is the or the non-empty list co-monad on sets. It's worth pointing out, so I didn't mention them, they just appeared on the slide, but there are these things called aerofoil-frase games, which capture the n-morphism relation. And we can equally have motivated this definition by looking at strategies in the aerofoil-frase game, so that's discussed in more detail in the paper. But it's nice to know that this is a kind of quite robust definition, because you can arrive at it in more than one way. So we can do a very similar thing when we also restrict the number of variables, we just have to carry along this restriction with us, but it's not too complicated. So how do we get, how do we pick out a structure whose canonical query can be written with most k variables, as well as quantifier n, well it needs to have this thing called an nk cover. So an nk cover, first of all you have an n-forest cover and then on top of it you have a labeling function. So if you want your, if you can only use k variables to talk about a structure, what happens when you relabel something is you forget the thing that you were talking about before. So this condition here I've written out in full, so if you have two elements who you want to be related somehow, then they should lie on the same branch because it's a forest cover and then one has to come before the other. So let's say a comes before b on the branch, then you can't repeat the label of a on the way to b, including at b. So that's kind of saying, well how do you, how do you then get this formula? Well you just take the label to be which, which variable you quantify over to talk about the element. And the least k such that you can find an nk cover for any n is called tree width and that's a very important wealth state parameter as well. So if we just take a little example of that, so this is going to need at least three labels to, so we had this forest cover from before and this will need at least three labels to forest cover it. So let's see why. Well if you imagine, if I take a look at d, so d is going to have a label, let's say it's got a label one. d and c have, there's an edge between d and c, so that means that d and c have to have, d and the label of d cannot be repeated. So b and c must have different labels to d, but there's also an edge bc, so that means b and c have to have different labels. So between db and c, I need at least three labels. It's worth noting that if the, the number of labels that you need is always at most the depth of the forest. So we can get a co-monad and we'll call that pnk in a very similar way. So we get one arising out of some junction between this category of pairs of structures and nk covers and then everything runs exactly as it did before. So the co-clisely category will give us a compositional structure behind this nk morphism. We can define nk queries and so on and nk covers will tell us about tree width I suppose as well as tree depth if you ignore k and tree width if you ignore n. And there's a corresponding game called the n round k pebble game and kind of as I noted you can always, if you don't care that much about k so you just want to get en back, then you can actually use pnn because it's not a restriction to be able to use n labels to label a tree of depth n. So the sort of concrete description is very similar to before but we need to carry around the labels of our elements within pnn. So the universe is sequences of length at most n of pairs a plus a label for a. When you define relations so you need the following two relations, two conditions as was there previously and then you need to say that if si is a subsequence of sj then the label which is I've denoted here pi a does not occur again in the sort of suffix of si and sj. So sj is remember a super sequence of si so you can't use the label of si again in t which is the suffix. So the the action on morphisms, the co-incommonplication etc are very similar to before you just carry on so this is the only place really where we say anything about the labels other than that they just sort of carried along and we will comment again that we can arrive at this definition as well by looking at strategies in the n round k pebble game much like we could look at strategies in the n round for a phrase again to get this co-monad so again this has this sort of robustness of having multiple ways to arrive at its definition. So I've just sort of summarized both on this page so you might like to pause it if I was going way too fast which I probably was. So this is what's nice here is that this kind of drives home the point is what we're doing is looking at things in finite model theory and then tidying it up using ideas of semantic ideas coming from category theory i.e our whole picture we can condense into the features of a single categorical construction which is this co-monad so it tells us about these n morphisms it tells us about these forest covers and it tells us about tree depth all in one. So I mentioned at the beginning we would talk about formulas with free variables and associated structures with distinguished tuples and you can run exactly the same story so I could have just done it with this along the way but it probably would have been more irritating just to carry them along. So all we really need to know is what are n forest covers and nk covers in order to make all of our definitions. So an nk cover of a with a distinguished tuple a bar is just an n plus l k cover of a but with the it's a special cover of a where you have to assume that you've put a one up to a l at the top of the forest so I suppose they'll make it into a tree because they're all at the top of the forest and you just color a well you label a one with one and a l with l. So yeah what's kind of going on there is everything should be able to see a one up to a l so you can think about a formula with free variables as kind of being like a formula with not free variables but you just quantified over those at the beginning and therefore everything should be in the scope of those and then the corresponding co monad we get we've used the same label for because it's identical pretty much is then going to be the the sub co monad or the sub set of p n plus l k um no sorry the definition's a bit wrong there so that should say p n k and this should say p n plus l k and it's all those sub sequences which start with this sequence um a one one up to a l or they don't necessarily start it could be a subsequence of that so they are the start they're comparable with that um and you can do the same thing for e n so I just did the slightly more uh complicated one which was p n okay so now we'll move on to talk briefly about the homomorphism preservation theorems there's going to be less detail in this section because there is a well there's a lot of detail on it in the paper um because it's sort of technical and fiddly so we're going to try and avoid talking about the technical and fiddly stuff um so what is the sort of the the basic homomorphism preservation theorem um so it's a classic uh theorem of model theory uh is that if you have a formula preserved under homomorphisms so that is to say if a models a formula um and there's a homomorphism from a to another structure b then b must also model that formula and I can find some existential positive phi which also models um or which is equivalent to sorry I said phi existential positive psi there's equivalent to phi uh rosman improved this um by by using uh or by even keep the quantifier rank constant so if I knew that the quantifier rank of my formula that I started with which was preserved under homomorphisms was n then the quantifier rank of the formula that I end up with um is also at most n so the you prove this as you might expect I suppose by you know you use a lot everywhere all of the definitions and theory from the last section um so we can reformulate the proof of it um phrasing things in terms of the encomonad rather than those sort of raw ideas in a slightly more uh categorical way I suppose um so that leads us to the following conjecture so this was from bransky just the the idea is pretty simple as well we we've got this theorem we proved using the encomonad um and we have this other comonad which looks very similar so uh can we just replace en by p and k and get the uh equi rank variable homomorphism preservation that I shouldn't say hp t I suppose it should just say hp um conjecture and uh which just says you can preserve as well as the the quantifier rank the number of variables um and get the same thing and so no this is currently still a sort of conjecture in the general case so we can prove it in the case where n is not that much bigger than k um so remember if n equals k that's just the equi uh equi rank theorem when n equals k so if n is a little bit bigger than k we can prove it so it's a slight refinement uh but not by much uh so the idea is can we just replace it and the answer is not in general so my story beforehand was that en and p and k share lots of similar properties um and in investigating this we found one that they didn't share um uh which is unfortunate because then we couldn't just carry over the proof uh so namely that is that uh en distributes over co-products in c sigma l so that was why I uh introduced how en uh acted on c sigma l so I can talk about this um so that's just to say in particular that I can find amorphism from en of a plus b to en of a plus en of b so just as a sort of heads up how what are these co-products look like in the category c sigma l you kind of you take the disjoint union of a and b you then uh then you identify as you take a quotient uh and you identify the elements of the tuple of a with the elements of the tuple uh the corresponding elements of the tuple of b and then the relations you take by just literally taking the disjoint union of the of the relations so we could phrase that um in terms of just um en morphism so the kind of property that we want is that if there is an en morphism from a to c and an en morphism from b to c then there should be an en morphism from a a plus b to c um but this fails in general um in c sigma l if n is anywhere bigger than k pretty much so if n is bigger than k plus one this fails um so just have a little look at the proof of how this um fact works um and what do you do you take so you take your so I want a homomorphism from en a plus b to en a plus en b um so I take an element of it uh and if so this is a horrible notation but it is sort of the correct notation um so if the last element of that sequence is in a but not um not one of the distinguished elements um then I just delete all of the elements of b to get a sequence of a um and if it was one of the distinguished elements um then I just set f of s to be um the sequence the subsequence um a one up to ai so why that kind of works is that this in this case it's kind of obvious why it works I suppose the fact that a and b don't interact so I'm going to preserve last elements of sequences I'm going to preserve subsequences um and that's exactly what you need to to be a homomorphism if you if you go back and double check the definition of en a um and then this will also preserve last elements and it's also going to preserve comparability of sequences precisely because uh all of the sequences were in the definition of um en a on uh en on the category with distinguished tubules or assumed to start be comparable to these sequences anyway um but so what's going to go wrong when we use p n k is there was that third condition in the definition is we needed to preserve comparability of sequences we needed to preserve last elements and then we also couldn't overwrite labels so we don't really know what to do when we're looking at the distinguished elements um and when we try and do this for p n k and in fact this just doesn't work so we'll just give a counter example in a second so here we go um so here's my counter example so this top one is my structure a so if i have some relation r and it's just this three cycle and then i have this seven length loop um so i'm just going to move this i have this seven length uh seven length loop um so what's going on is um well sorry i have my seven length path and then i have just a unary relation on b4 and if i take the co-products of a with distinguished element a1 and b with distinguished element b4 then i glue this onto this along a it's glued on to b um and the property is that a should now have a kind of path of three elements so the the resulting co-product will have a path of three elements such that the first element satisfies you and the second element satisfies you um so that will that will be satisfied by a plus b but both a and b uh map onto b and uh but then a b doesn't have this property that there is a path of three things with the first element satisfying you and the second element satisfying you so there's no way b um this will be a formula that a sat a plus b satisfies and b does not so that's you you just can't have this morphism in general that i claimed uh so let me move this back again um right so we're going to just give a very short sketch of the proof of the equi rank hbt um so we can reduce it pretty quickly um to a slightly easier thing to prove um so a formula is preserved under n morphisms uh implies that is equivalent to a positive existential formula so how do you do that you literally just take the disjunction uh of um all of the n canonical queries and again we're relying on the fact that there are only finitely many positive existential formulas so this is a finite disjunction and it's pretty easy to check the satisfy this well this formula is well it's pretty trivial that it has quantifier rank at most n and uh will be equivalent to um theta precisely because theta is preserved under n morphisms um and then using the comb on it we can kind of reduce this one step further because we can factor um uh general n morphism into uh the the special n morphism from a into e n of a and then um a homomorphism so we would all we need to prove now is that any formula preserved under homomorphisms is preserved under this particular n morphism from a into e m a b j um so we'll just give another one more definition uh so a this is a grading elementary equivalence so a is equivalent to m b if uh for every formula in l n a believes phi if and only if b believes phi so this is not just positive existential ones anymore um and why is this what uh we define this just to say this is now the um more thing another thing that we can sort of transport um formulas in l m through because obviously any formula or bounded quantifier rank is preserved under this thing and um so just we can use that to to restate the hp t at the bottom there um so because any formula of quantifier rank n is preserved under this thing so being we now need to prove that being preserved under this relation and homomorphisms implies being preserved under that and that will be sufficient um and just a few notes about this um why do we talk about it and why do we need this thing uh why do we need to consider extra um and why do we need to consider uh structures with distinguished elements when we you know we're not working in those categories at all um and the answer is it's to build these um partial isomorphisms uh which can be used to characterize uh this relation so this can be characterized by another version of the aerofoid phrase a game where you extend sequences of partial isomorphisms um and this this relation which should have an n here sorry um so if you have a morphism an n morphism one way and back then that implies a partial isomorphism between the distinguished elements which is what we use to build on so we use companion structures um in order to try and do this so if we want to prove being preserved under this move from a to e na is um implied by being preserved under morphisms and really what we need to do is you know chase some diagrams so if we can find an a tilde and e na tilde uh then that will be good enough so we could just chase theta around there because preserved under homomorphisms and then through here because it has quantifier at most n and then down again uh so i'm just going to give a very brief intuition on how you construct it because the the sort of quite detailed um so the idea is to build a superstructure of a which we'll call uh sigma uh well it's called a tilde uh which is very good at extending partial isomorphisms because then we'll be able to find this relation much easier um and we do that by taking an increasing sequence of structures sigma i and at each step we add elements from it in order to be able to extend partial isomorphisms that are involved with that uh or involve sigma i of a so we add elements so that it becomes better um at extending partial isomorphisms um and then if we take a co-limit all of those things uh it will be able to extend partial isomorphisms from itself essentially uh rather than just for the previous um sequence structure in the chain and so i've just included a note here that we do in fact use these co-products a lot um so i mean again these aren't structures in c sigma so we need to use this co-product over c sigma l um and then forget it back down into into c sigma um and e na is constructed in an identical way um and again they're both very good at extending partial isomorphisms because they have a sort of basic relationship between them that implies this um and elementary equivalence up here to complete the diagram for us um so as i stated here this is where it's not going to work in general for for p n k so this only works when again n is not that much bigger than k uh so it's smaller than k plus two we managed to approve just by kind of fudging something through so our previous gen more sort of general lemma that we needed failed about distribution over co-products failed when n was bigger than k plus one but we can get one better um so this is just a few comments so uh this is quite a complicated construction and a reasonable question is you know it does weigh more than is necessary so for example there's symmetry between en a tilde and a tilde and that's completely unnecessary you know they're both satisfying more properties than they need to because they need to satisfy different properties and they satisfy all the same properties um so we ask like can you tweak it and it turns out to be quite difficult so it seems every time you try and change one thing to get one property that you need then you happen to kind of lose another one um but it's not uh it's not necessary that this should work so we have found a difference between the co-pliesely categories of en and the co-pliesely categories of pnk so it might be that that is sufficient to find a counter example or force it uh force it there so that's the end of my talk so thanks for watching um and yeah just to say stay tuned because there's hopefully going to be more work in this vein about co-monuts and co-monuts similar to these coming out in the future