 I am happy to introduce Jason Parker from Bandon University who is going to talk on the Covariant Isotropy of Gothic Toposis. So please go ahead. Great, so thank you for the opportunity. So I'm Jason Parker and just FYI, I do sometimes stutter or stammer, so you hear me stuttering or stammering, that's totally normal. So yeah, so I'm talking about Covariant Isotropy of Groten Dika Toposis. So I'll just start with some kind of background. Whoops, I have to get that going. Okay, yeah, so basically Covariant Isotropy is a somewhat recent categorical construction that basically gives an abstract notion of conjugation or inner automorphism for an arbitrary category. Oh, I think I have a, okay. So basically in prior work with Peter Prostra and Phil Scott, we use techniques from categorical logic to characterize the Covariant Isotropy group of any locally presentable category C and in particular of any pre-sheaf category. So in this talk, we'll first give an overview of Covariant Isotropy and then show that its characterization for any pre-sheaf category basically extends to any Groten Dika Toposis. And this is based on my recent pre-print with the same title as this talk that's currently on the archive. Okay, so just to first motivate this topic. So basically George Bergman proved in a certain paper from I think 2012 that the inner automorphisms groups can be characterized purely categorically as the group automorphisms that extend naturally along any group morphism out of the domain. So see this, first observe that if I have say an inner automorphism alpha of some group G induced by some element S in G then for any group morphism F from G to some further group H, we can push forward alpha along F to define a further inner automorphism alpha sub F of the codomain H simply by conjugation with F of S and H. So therefore in particular, if I push alpha along the identity on G, we just have alpha. Now this family of automorphisms that I obtained from the starting inner automorphism alpha is coherent in the sense that if I have any group of homomorphisms F from G to G prime and then F prime from G prime to G prime prime then the following square will always commute. So we have alpha sub F on the top, alpha sub F prime after F on the bottom and F prime on either side. So basically if I push forward these or the inner automorphism alpha along F and then along F prime after F then those kind of extended or those pushed forward inner automorphisms will be coherent in the sense. Okay, so for group G, that's called an arbitrary family of automorphisms with above naturality property an extended inner automorphism of G. So concretely, this is a family of group automorphisms, one for each group morphism out of G on the codomain of that morphism. And these are all coherent with each other. And now I don't assume a priori that these are induced by a starting inner automorphism of G. So this is just an arbitrary family of automorphisms on the codomains of all the group morphisms out of G that are coherent in the sense on the previous slide. So this is just a natural automorphism of the projection functor from the group G slice group to group. So Bourbon proved that if I have any automorphism alpha of a group G then alpha is an inner automorphism. So it's induced by conjugation with some elements. If and only if there is an extended inner automorphism in this above sense that induces alpha. So who is component at the identity on G is just alpha itself. So we just saw that if alpha is inner then it induces any such extended inner automorphism but the converse is also true that if I give you any arbitrary extended inner automorphism of a group G then it must come from an inner automorphism in the conjugation theoretic sense of the starting group G. So basically this gives a completely a categorical characterization of inner automorphisms of groups. They're exactly those group automorphisms that can be coherently or functorily extended along group homomorphisms out of the domain. Okay, so how does this relate to your kind of inspire covariant acidity? So we have a functor Z from a group to group that sends any group G to its group of extended inner automorphisms. And these do form a group. So we refer to Z as the covariant isotropy group functor of the category group. And actually Bergman's theorem shows that this is in fact isomorphic to the identity functor on the category group. So in fact, any arbitrary category C has its own covariant isotropy group functor ZC that sends each object C to its own group of extended inner automorphisms. So these are families of automorphisms, one automorphism on the codomain of each arrow out of the object that are coherent with each other in the sense as described on a prior slide. So these are just natural automorphisms of the projection functor from the slice category under the object to the category. So basically again, these are just families of automorphisms, one automorphism on the codomain of each arrow out of the object that are natural or coherent. So we can turn Bergman's characterization of inner automorphisms in group into a definition of the notion of inner automorphism in an arbitrary category C. So if we have an object C and an automorphism alpha, of C, then we can define alpha to be inner just if it's induced by an extended inner automorphism. So if there is an extended inner automorphism whose component at the identity on C is alpha. So basically this is saying that alpha is inner just if it can be coherently extended along any arrow out of the object. So group is the category of models of an algebraic or an equational theory so just a set of equational axioms between terms in the sense of categorical logic. So in prior work, we basically generalized ideas from the proof of this result of Bergman to give a logical characterization of the inner and extended inner automorphisms in this more categorical sense of the category of models of any finite area quasi-equational theory T. Okay. So, yes, okay. Oops, there we go. Okay, so just I'll just get some background on those quickly. So a finite area quasi-equational theory T in the sense of the paper by Palmgren and Vickers called, I think, partial horn logic and Cartesian categories. So it's over a multi-sorted finite area equational signature. So it has just sorts and operation symbols. So it's a set of implications called the axioms which are between funnitary horn formulas which are just conjunctions of equations between the terms. Now, the kind of relevant feature of these theories and these are also known as say Cartesian theories or essentially algebraic theories is that the operations symbols may only be partially defined. So we have a term T, we'll write T down arrow as an abbreviation for T equals T meaning that T is defined. So this is not always a theorem of the given underlying logic. So now, if we have any regular cardinal lambda then one can define an extension of these theories to lambda. So we can define say a lambda area quasi-equational as the theory T in the same way but now allowing for operations with possibly lambda arity and then also lambda area conjunctions. Okay, so just to give some examples of these. So any algebraic theory is a funnitary quasi-equational theory as are the theories of categories, group points, stripped monoidal categories and any pre-sheaf category. Now, for this talk, so if we have a small site, then the Rodin de Topos on that site is in fact the category of models for a lambda area quasi-equational theory where lambda is the smallest regular cardinal larger than any of the sips. So the sorts of the objects of C for any arrow F I have a unary operation symbol going in the opposite direction and for any covering sieve J we have a lambda area operations symbol sigma J of that typing. And then you have axioms expressing the contravariant functoriality and the fact that any matching family has a unique amalgamation. So therefore the models of this theory are exactly the sheaves on the site. Okay, so now I will just kind of briefly give an overview of the main results for covariant isotropy of Cartesian or quasi-equational theories. So it's fixed a lambda area quasi-equational theory over a lambda area signature and we have its category of models T mod. So again, I'll just review the characterization of its covariant isotropy group which was achieved for funnitary theories in prior work and extended to lambda area in my recent pre-print. So basically using the quasi-equational syntax of T one can define a notion of definable automorphism for a model M of T and these then form a group, a depth in of M. So if T is a single sorted for simplicity then given any model M one can form the T model M bracket X obtained from M by freely adjoining an indeterminate element X. So the elements of this model can be seen as congruence classes of terms involving X regarded as a new constant and also constants from the model M and here two terms are congruent if they are provably equal in the diagram theory of the model extended by this axiom saying that this new constant is defined. So the diagram theory of M extends T by basically adding constants from the model M and axioms expressing the relations that hold in M. Okay, so if we have some element bracket T and M bracket X we say it's substitutionally invertible if there are some possibly other elements bracket S in this model that is basically the substitutional inverse of T with respect to this diagram theory plus X. So we substitute S for X in T we get just X in the diagram theory and conversely. If we have say an operation symbol of Sigma then a given element bracket T and M bracket X commutes generically with F if T of M with X1 through XN proves the sequence. So first of all, this model is just the diagram theory extended by N new pairwise distinct constants which are all defined. And then the sequence basically is expressing that if F applied to this N tuple of new constants is defined then if I substitute it for X in T that is equal to F applied to these sort of N variants of T. So basically this is supposed to express that T is commuting or that T commutes in a sort of generic sense with this operation symbol F. And likewise, this congruence class bracket T will reflect the finalness of F if the same theory proves this sequence which basically says that if T with F of this tuple for X is defined and so is F of this tuple. So this is again expressing that this congruence class bracket T reflects defining this of F in this generic sense. Okay, so we can then define a group. So def N of M to be the group of all of these elements bracket T in M bracket X that are substitutionally invertible and commute generically with and reflect defining this of every operation symbol of the signature sigma. So basically this is the group of definable automorphisms of M where we regard an element bracket T in this in this T model as inducing a definable automorphism if it satisfies these properties. Okay, and if T is multi sorted one can extend the above the definitions appropriately. So basically you just have to of course account for the possibly many sorts which is not too hard to do. And that is shown in certain of the references. Okay, so then we proved in prior work that if we have a lambda area quasi-equational theory then for any model M of this theory it's covariant isotope group. So it's group of extended inner automorphisms is actually isomorphic to its group of definable inner automorphisms. So, and then in prior work namely in this initial paper isotope of algebraic theories we use this result to show that the sort of inner automorphisms categorically defined in many categories of algebraic structures like monoids, groups and abelian groups non-commutative rings, et cetera are exactly the sort of conjugation theoretic inner automorphisms that you would expect. So in this sense, covariant isotope does actually provide kind of a generalised notion of inner automorphism for a category. Checking the time, okay. So for pre-sheet categories so in a recent paper we also characterized the covariant isotope group of any pre-sheet category for of course a small category C. So we have a pre-sheet F and we show that the group of definable automorphisms of F consists up to isomorphism of exactly the natural automorphisms alpha of F induced by some element psi in the automorphism group of the identity functor on C. So here psi is a natural automorphism of the identity functor on C. And to say that alpha is induced by psi means that the component of alpha at any object of C is just the functor F applied to the component of psi at C. So basically the, this shows that the only inner automorphisms of pre-sheets are those induced by natural automorphism of the identity functor on the base category. So then follows that the covariant isotope group functor of any pre-sheet category is actually a constant on the automorphism group of the identity functor of the index category. Okay, so now in my recent preprint I wanted to see if this would extend to arbitrary growth and deptopuses or sheaf categories. So to start, for certain convenient technical reasons I first just examined sub-canonical sites where no object is covered by the empty sieve. So basically for any object C, the empty sieve is not a covering sieve of C. Okay, so then if F is any sheaf over such a site, I show that its group of definable inner automorphisms consists up to isomorphism of exactly those natural automorphisms alpha of F induced by some natural automorphism psi of the identity functor on C in the sense I described on the prior slide. So now this proof is quite technical and it's the most non-trivial aspect of the overall results in that preprint. So although this result is the same as for pre-sheafs, so far at least, its proof is still not obvious or non-trivial and so, yeah. Yeah, so as for pre-sheaf categories, if CJ is any small sub-canonical site in which no object is covered by the empty sieve, then the covariant STOB group of that sheaf category is constant on the automorphism group of the identity functor of the base category. So now we want to hopefully remove the assumptions of sub-canonicity and no object of being covered by the empty sieve. So the second property is easier to dispense with. So if we have any small sub-canonical site which doesn't necessarily have this property, one can find pretty easily another sub-canonical site which does have this property, so in which no object is covered by the empty sieve and where the sheaf categories are equivalent and the automorphism groups of the respective identity functors are isomorphic. So therefore, if CJ is any small sub-canonical site where objects may or may not be covered by the empty sieve, then the covariant isotropy group of the sheaf category is still constant on the automorphism group of the identity functor on the base category. So now we want to consider any small site not necessarily sub-canonical. So first, if we have any locally small category E with a small full dense sub-category C then I just proved this lemma saying that the automorphism groups of the identity functors on those two categories are isomorphic. So, and I use that to prove the following results. Well, actually, okay. So if we have any small site CJ, not necessarily sub-canonical, then it's well known that there is a sub-canonical topology K on the small full dense sub-category AYCs. That's just the full sub-category of the sheafifications of the representables for which the original topos is equivalent to the topos of sheaves on this site. So then the covariant isotropy group of the original sheaf topos is constant on the automorphism group of the identity functor on this full sub-category AYC. And by the lemma in the first point that's in turn isomorphic to the automorphism group of the identity functor on the actual topos. So basically, if we have any small site, not necessarily sub-canonical or and where objects could be covered by the empty sieve, then the covariant isotropy group of this topos is constant on the automorphism group of the identity functor of the topos, which may also be called the center of the topos. Okay, so in particular, since of course any pre-sheaf category is known to be the category of sheaves for the trivial and sub-canonical topology T on C where only maximal sieves cover, we recover our early result for pre-sheaf toposes. And now if the site is not sub-canonical, there is in general no relation between the automorphism groups of idC and the identity functor of the topos. So for example, if the automorphism group of idC is non-trivial, say for example, if C is the one object category corresponding to an abelian group, which is non-trivial, then, and J is the topology where every sieve covers, then it's easy to show that J is not sub-canonical and that the only sheaf is the terminal pre-sheaf, so the sheaf category is trivial, so that's the automorphism group of the topos is trivial even though the automorphism group of idC is non-trivial by assumption. And yeah, so our results shows a major difference between covariant and contravariant isotropy of Groedendie toposes, so the latter was studied in this paper called isotropy and crosotobuses by Fung, Kostra, and Steinberg, and they show that so contravariant isotropy is always representable by a sheaf of groups, while we have now shown that a covariant isotropy is always constant, and in fact, on the group of global sections of this sheaf of groups. Okay, so to start wrapping up, so via George Bergman's purely categorical characterization of the inner automorphisms of groups, covariant isotropy can be seen as giving a notion of conjugation or inner automorphism for arbitrary categories. So basically you can define an automorphism in an arbitrary category to be inner if there is an extended inner automorphism of the object that induces that initial automorphism or in other words, the automorphism is inner if it can be coherently extended along any map out of the object because this is the result that holds for, that characterizes the inner automorphisms of groups defined in terms of conjugation. So we've characterized the covariant isotropy group functor of T-mod for any lambda, airy quasi-equational theory T. So the covariant isotropy group of any model M or its group of extended inner automorphisms is isomorphic to the group of definable automorphisms of M. And this, I think that this characterization is most useful because it can provide a more kind of concrete or perhaps computational way to characterize the extended inner automorphisms of a given model because those, I mean, the definition of extended inner automorphism does not kind of in itself allow for any obvious way to characterize them. Whereas if you work with the syntax of a theory for which this category is the category of models then via this isomorphism that kind of provides a more perhaps down-to-earth computational concrete way to hopefully characterize the covariant isotropy of the category. Yeah, so using this results, we've shown that the characterization of covariant isotropy for pre-sheet topuses that we achieved earlier, in fact, essentially extends to all Rotendieke topuses. So for any small and sub-canonical site, CJ, the covariant isotropy group of the Rotendieke topos is constant on the identity functor on C. And as we saw before, this basically amounts to saying that the only inner automorphisms of a sheaf on any such site are the ones induced by a natural automorphism of itsy. So although this result shows that covariant isotropy as opposed to contravariant for Rotendieke topuses is in some sense degenerates or at least constant, the proof of this is still non-trivial. And we also intend to build on this result to characterize the covariant isotropy of categories of sheaves of algebraic structures, not just sets, which will be non-constant in general. Since in my thesis, I showed that the covariant isotropy of categories of pre-sheaves of algebraic structures is non-constant in general. Okay, so that concludes my talk. Thanks very much. Oh, and have some references here as well. Thank you very much for your talk, Jan.