 Joshua Brickley from the University of Insovria will talk about the logic and geometry of localic morphisms. Thank you very much to the organizers for allowing me to speak of this presentation. The story I want to share with you today is about how we can think about topuses as ways of completing a structure. So we might start off with some structure which doesn't have all of the properties we want it to have and topuses are going to give us a way of talking about what structure there should be there and what what structure we can add. And so in particular, as the title says, we'll be looking at how we can add structure to some certain logics coming from localic morphisms. So the idea behind the presentation I'm going to show you is that we want to utilize our intuition coming from a propositional setting. So in a propositional setting we have localic topuses classifying propositional theories, but we also have localic topuses obviously coming from locales. And so what we want to be able to do is extend this to where we talk in a relative toposclerotic sense about having a localic morphism over a topos and that corresponding to an internal locale. And then on the theory side we want to be able to talk about what sort of geometric theory we might get out of that. So that's the overall picture. I'm going to give some idea of what's going on with these branches of the bridge and then I'll give a specific case which ties in very nicely with the logic. So yes for people who haven't seen the definition before, a localic geometric morphism is a geometric morphism from a topos F to a topos E such that every object in our domain topos is a sub quotient of something in the inverse image of A. So some examples of localic toposes. Every inclusion is localic. That should immediately tell us that every topos comes attached with at least one localic morphism to another one because every topos has an inclusion into a pre-shift topos. And we can also get lots of examples of localic morphisms coming from locales. So when we take sheaves on a locale, this obviously gives us a localic topos, but every arrow between two localic toposes is also going to be a localic geometric morphism. And we can do the same with internal locales and do exactly the same construction relatively. So here we take the locales internal to a topos and we take sheaves, internal sheaves of these locales, and this gives us an embedding of the internal locales into constant toposes that has precisely the same properties that you'd expect from the normal localic setting. And every arrow between these localic toposes is going to be localic as well, although there's a little caveat to that because obviously when we take nevus embedding into here, there comes a choice of the morphism to E. And so it will be localic up to that choice of arrow. And so we also have, by a result from I believe Joao and Tierney, that every localic geometric morphism is going to be, the domain topos is going to be of the form given here that F is going to be the internal sheaves on the internal locale given by the direct image of a sub-object classifier. So this result gives us a very nice description of what the internal locales should be corresponding to a localic geometric morphism. And so this is an example of localic geometric morphism. So what is an internal locale? For the application I'm going to give at the end, we're only going to be concerned with internal locale's pre-sheaf toposes. So an internal locale of the pre-sheaf topos, where C has finite limits, is going to be a functor from C into the category of locale such that all of our arrows have a left adjoint satisfying Frobenius reciprocity. This condition is Frobenius reciprocity and also we have the Beck-Chevrolet condition holding. This, these should be F's down here, I forgot to change that. So those are internal locale's and so now we can appreciate the arches of the bridge here that the the localic morphism is going to go to the internal locale given by the direct image of a sub-object classifier. And similarly internal locale's are going to induce localic morphisms by the projection into C. So now I want to move more onto the logic away from internal locale's but we'll still see lots of localic properties cropping up. And that is in the idea of a localic expansion of a theory. So a localic expansion is where we take a theory, we take its signature and we add new relation and function symbols to that signature and then we take a theory that proves all of the axioms of our previous theory. And now it's a theorem due to Caramella in theory sites and toposes that every localic expansion produces a localic morphism between the classifying toposes. And so we can make the observation that every theory in a signature sigma is going to be a localic expansion of the theory of objects without signature. This is the empty theory in the signature containing just the sorts and the signature of our theory. So no function symbols, no relation symbols, just a quality. And so we're going to have that every classifying topos is going to be localic over the classifying topos of the theory of objects for its signature. So the theory of objects is we're going to build a explicit site for it and it's going to involve the category of relavings of sorts, which I've denoted as sort subscript sigma. Now this category has as objects the finite strings of variables in the in the sorts. And then arrows between these finite strings of variables are going to be relabelings of variables. So that means they are maps that respect sorts. Now obviously if sigma is single sorted, then a finite string of variables is going to correspond to a finite set and any relabeling because every map is going to respect sorts is just going to be every map. And so we have the category of relayed lengths for a single sorted theory is given by fin sets. And so then we can observe since the classifying topos of the theory of objects in the signature sigma is going to be isomorphic to this pre-sheaf topos, we're going to have that every classifying topos is the topos of sheaves on an internal locale of this pre-sheaf topos. So that's that's the sort of motivation for what we're going to do. Next we've seen that whenever we have something that's locale over another topos, we're going to get an internal locale. And here we're going to see that we actually get quite a nice description for the internal locale corresponding to a classifying topos of a theory T. Now obviously since every gottendic topos is a classifying topos of some theory, we're going to have that we're able to perform this construction for every topos. So let's just review our bridge in this restricted context. We have as the unifying notion topos is being localic over this pre-sheaf topos and this is going to correspond to two notions on the one hand the geometric notion of internal locale this pre-sheaf topos and on the other hand the notion of geometric theories in a signature sigma. So that's that's we already appreciate what's going on with this bridge but we need to sort of understand this bridge as well and how they interact. So yeah in order to do that I'm just going to quickly recap some of the stuff from LeForg's lecture earlier. It's not last week you know it is last week it's not this week so at the end of last week I'm going to recall some of the notions from syntactic sites for a geometric theory. So for a geometric theory T we can take the syntactic category which has its objects but formulae in that context and the arrows between two formulae in context are going to be provable equivalence classes of formulae theta such that and then we have these sequence holding and these sequence as LeForg explained just express the fact that theta is the graph of a function. So that's that's the usual definition of a syntactic category and its syntactic topology has this description but to SID is covering if and only if we have this dis disjunction of existential statements being satisfied. And then it's it's very well known that the classifying topos of a theory T is going to be the sheaves on this site with this topology and it's also going to be important later but we'll see we'll see how we can reconstruct it but the topology is sub canonical. So that's the syntactic site but and so it's going to be the classifying topos is going to be localic over the pre-sheaf topos involving the sorts on the signature and so we need to describe what the sort of internal locale is going to look like and for that I'm going to introduce this notion of a substitutive syntactic sites. So here given our geometric theory we're going to give it an internal locale of the pre-sheaf topos so it's going to be a functor from the opposite of sorts to locales and we're going to associate with every context every string of variables x we're going to associate it the locale given by the formulae in that context ordered by syntactic proof and whenever we have a relabeling we're going to send that to the locale with corresponding frame homomorphism given by relabeling under the aneurysm sigma. So here in this notation the frame homomorphism corresponding to us locale is going to send formulae psi in context y to the formulae in context x where we substitute every instance of the variable y by sorry every instance of say like yi in this vector y we're going to substitute for yi sigma of yi which is now going to be a variable in this context and since we assume all of our contexts to be disjoint there's no issues with that there's no issues in the order in which we relabel the variables so this is going to be our associated internal locale how do we know that it's an internal locale it's an internal locale because of this pre-sheaf topos and the Beck-Chevrolet and Frobenius conditions basically are saying that existential quantification behaves well with substitution they're quite easy to show and the second remark I want to make is that the left and right adjoints oh I mean that this this is a wide Beck-Chevrolet and Frobenius hold is that the left and right adjoints of our frame homomorphism are given by existential quantification and universal quantification so this is the yeah and so we have this internal locale and we're now going to take internal sheaves on that and we're hoping that the internal sheaves are going to be closely related to the classifying topos and indeed we're going to see that they are so yes when we take the relative construction when we take the Grottendijk construction of this internal locale which if you you can refer back to Olivia's lectures for the definition of this category but in this in this particular instance we get a very nice description that it's going to be objects, formulae and context and the arrows in this category are going to be relabelings of variables yeah so if we've got a an arrow between phi in context x and psi in context y is going to be a relabeling in the other directions such that we have this sequent holding in our theory and then when we take the induced apology because every internal locale has a induced apology coming with it we end up with a family of arrows in this a family of arrows in this category being covering if and only if we have we have this sequent holding and so because this is the description of the Grottendijk construction and the induced apology we have that the sheaves on the on this internal locale is going to be equal to the sheaves on this site and so if we look at this it looks very much like the usual syntactic category just that we've got we've got the same objects but we've got a lot fewer arrows and sometimes our arrows will be identified oh and I should also mention that that this topologies are generated by two species of covering families and so this this corresponds to the fact that we've got a horizontal and vertical covering data coming in from our relative topography and so yes so indeed we end up with this site being a dense sorry having a dense morphism of sites to the usual syntactic site and the the morphism of sites is going to be given by this functor eta which sends a formula in context to itself and it's going to send a relabeling to this provably functional formula here so I'm claiming that this is a dense morphism of sites and so if that's the case then as a corollary we get that this is a alternative description for the site of a classified topos of the theory T so just to recall what a dense morphism of sites is it's a function sorry a functor from c to d satisfying these conditions s is j covering if and only if it's images k covering for every d we have a k covering family of morphisms 2d and whenever we have an arrow between f of c1 and f of c2 that you know it might not necessarily come in the image of f then we want to show that it can be uh it's it's densely generated by arrows in the image of f and um whenever we have uh two arrows being identified this is the fourth condition um then whenever we have two arrows being identified by f then there exists a j covering family of arrows uh which uh exhibit their equality um and so when we have a dense morphism of sites then we've got an equivalence of categories so let's show that eta is indeed a dense morphism of sites uh so the first one is immediate by definition because the uh the groten-dict apologies are so similar um the second condition that uh it has a covering siv onto every object follows immediately because it's subjective on objects it's the next to which uh less immediate um so let's suppose we have an arrow coming from a provably functional formula in our syntactic category which might not necessarily just be a relabeling of variables um we're going to consider the diagram here um where we uh take the formula witnessing this uh provably functional formula um and now we uh have due to uh all of the um due to all of the sequence that a provably functional formula has to satisfy we have an arrow going to x phi and it's composite we can form the composite as LeVogue showed us in the previous series of lectures um thusly and now uh we just need to show that um we need to show that uh this is uh coming from our substitutive syntactic site is dense and this is also coming from our substitutive syntactic site um but this is immediate uh uh from all of the extra sequence that provably functional formulae have to satisfy uh so this arrow is covering because we have this sequence holding and this arrow is uh um in the image of eta by the equivalence of these two formulae um and for the final condition if we've got uh two relabelings um such that their eta associates the two of them then we're going to have that phi proves that sigma of y i is equal to tau y i for each y i uh in our context um and so now we need to find a series of we need to find a covering family uh that witnesses this equality but this comes from this commutative diagram here where uh this is the co-equalizer because of course all of these relabelings go in the opposite direction this is the co-equalizer um of these two relabelings and uh this is this is the projection this is the map given by projection from this to the co-equalizer and we want to show that it's a covering arrow um but the required sequence namely this uh can be proved easily from uh the fact that uh since these two arrows up here are identified that we have this sequence holding for every y i in our context so we do indeed have a in conclusion we do have indeed have a dense morphism of sites um which is uh nice because now we we we have a nice way of talking about what the uh syntactic site is in terms of a localic morphism and so just quickly um I want to give some uh verifications of previously known results um so for example if we've got uh these are results from theory sites and toposes if we've got t-dash being a localic expansion of another theory t then these are both localic expansions of the uh these are both localic expansions of the empty theory on the source of the signature and so being a localic expansion um we end up with a morphism of locales which we can uh construct uh very in a hands-on way uh we get a morphism of locales which is internal to the pre-sheaf topos sorts on sigma interstates and so because we've got a morphism of internal locales we end up with a localic morphism between the two toposes and so we we uh verify our motivating result uh from earlier but um potentially more interestingly a quotient theory t-dash of another theory t is going to be a theory in the yeah this is a definition is a theory in the same signature um that proves all of the axioms of t and now we're able to uh sort of recognize the um other uh headline result from theory sites and toposes but there's a correspondence between quotient theories and subtopoi and uh the i mean that this that there's there's more work going on here than is immediately apparent on the page but we can appreciate the uh where the correspondence is coming from the subtopoi of a classified topos are going to uh correspond to inclusions of internal sublacals which in turn are going to correspond to quotient theories um and another result uh you uh which was i believe uh joeyal um well first of all it appears in joeyal and terny and secondly joeyal mentioned it at his uh talk at the toposes at ih yes that you can since we've uh since we have that every theory is localic over the um this pre-chief topos here we can use the fact that every theory is marita equivalent to the uh to a theory with a single sort to get that every theory is localic over the object classifier um which which is uh a very nice result that yes as mentioned originally appeared in joeyal and terny um and so finally i want to you know go all the way back to the beginning where i mentioned that we can think about toposes as telling us the information that we're missing out um and so we we had this substitutive syntactic site now where do the um the provably functional formulae come in uh they come in when we take a uh syntact yes they come in here where the syntactic category is going to be the full subcategory of the representables in the classifying topos um and then additionally the syntactic topology is going to be the restriction of the canonical topology so the way to see this is that uh because the topology is sub-canonical CT has to be a full subcategory and indeed it has to have the same objects as the representables so this is where our extra arrows are being added they're being added on the topos theoretic level um and yeah indeed we could also go the other way and uh if we didn't know that the topology was sub-canonical we could define the syntactic category as being the full subcategory of representables and then we would have that the uh induced topology the syntactic topology would then have to be sub-canonical um and yeah and then in Olivia's denseness um on a graph there's a description of how we can do this in an elementary way um so yeah thank you for listening that's my talk up thank you very much