 the talk is going to be about elementary higher toposes having a natural number of please all right and so thank you to organizers for giving me a chance to speak and thank you all for being here so this is a talk on like an ongoing project on studying so toposteoretic and specific like elementary toposteoric phenomena in like the higher categorical setting and there's something I specifically like about this title because unlike many foundation stocks I give this one already has like a specific sentence like the title already tells you what the talk is going to be about there's an object and has a certain property so the talk is obviously not going to be very long it's 20-25 minutes and so I have to skip some of the details of the proofs that I would have liked to give and so if there's anything that's interesting to you then please feel free to reach out to me and can talk me whatever I would love to talk more about this but I also want to respect the time here of this conference I know it's like a long day all right so let's go into the toposteoretic part so this should be familiar more or less to most of you can like break down toposes in various ways and this conference has been a great example of like the different kind of perspectives on the topos but like so one way to think about toposes is like there's like a gluttonyck perspective that arose like in the study of like algebraic geometry in approval key setting and then there's like an elementary perspective so the elementary toposes which is like maybe more connected to like type theory other stuff and obviously connected but also have like the differences and so gluttonyck toposes are in some sense like very nice they have like lots and lots of amazing properties they're like locally cottage and close then particularly presentable and they have subalgae classifiers and so we can use them to do lots of math unfortunately elementary toposes are not as nice so they did lose some of these properties so in particular elementary topos is by default not locally presentable and we still have some limiting elements like only the finite ones we don't have infinite elements and we only preserve some of the like the like axioms that you have in Gero's theorem so for example um co-products are still disjoint but not all the properties survive and so the ones that I have bolded here are the ones that we take as a definition right so we want an elementary topos to be a category that has finite limits locally cottage and close and has subalgae classifier and so with any generalization the benefit of it is that we gain more examples usually which for example the category of finite sets is an elementary topos but it is certainly not good indeed because it does not have infinite co-limits and there are other kind of interesting examples such as filter products or realizability toposes that come up in kind of more foundations and logical frameworks but then the drawback of the more examples is that it's like expressive right so there's bunch of stuff that we cannot prove anyone and in some sense the way I think about it is there's like this intuitive sense of like something infinite and if something is infinite then it's harder to like construct it in the in an elementary setting whereas in good and equal possessive payoff we can use local presentability to get this and so one thing we want is something that intermediates so we want to strengthen the notion of elementary topos without going all the way like without actually going all the way to a good and equal which has local presentability and one way of doing that is by the notion of a natural number object which is an object we add to elementary topos to strengthen the theory so and because the talk is a lot about national object that may actually give you like definitions and there will be several and each one of them kind of hinges on focusing on certain aspects of natural numbers in the category of sets so one way I think about natural numbers is that okay it's an internet set which means it has like a an injective but not surjective self-injection which cannot happen for a finite set and so if you try to use that to form some kind of axiomatic definition you end up with a freight natural number object which is just an object along with the point and endomorphism so that we have these two following two column diagrams then another aspect of natural numbers is well induction so we know that natural numbers have like the inductive property from a piano axiom or we can make that into like some kind of axiomatic definition i.e. we have a natural numbers we have an object we have the point and we have this endomorphism which is like a successor map and now we say okay whenever we have a sub-object that is closed under zero and the successor then it's just the object itself which is precisely the category statement of induction if one is in there and if n implies n plus one then you're hoping and then the final version which is I think the one that most people might not seen before so it's the one that if you just google natural number object it's probably going to like show up first is the Navier natural number object and that just says okay it's an object along with a point and endomorphism that is initial with that data so whenever I have another object with some endomorphism u and a point b then I get a unique map and that unique map just defines a kind of recursive function which just applies u n times 3d object and so I have now three notions and all of these three capture a certain aspect of n and fortunately all of these three coincide inside the elementary topos and that is why we usually don't actually distinguish so if you're just reading like a paper they will not distinguish between these notions they will just say you have an elementary topos plus natural number object because whichever notion you pick is equivalent to the other two so that's what we usually don't really matter don't care about this distinction and now let me come back to my original promise so as soon as you have a natural number object in our elementary topos we can now prove new results that we couldn't do before and there are many examples of that I want to focus on one here as soon as our natural object we get free monoids and the equation for the free monoid is explicitly given by by this formula here where like n1 is called the universal finite corner and it's just constructed out of the product of the natural number object itself and so notice this here the existence of the natural number object is necessary right it's like actually it's the vision and necessary but let's not focus on that it is actually necessary in fact if it doesn't exist this would fail and an easy example is finite sets inside the category of finite sets I don't have free monoids even a free monoid on the point or something doesn't exist because it wants to be n itself and n is not a finite set so assuming an natural number object is necessary to get such results okay so before I move on to the more general high category stuff let me summarize what we did so we can define an elementary topos we want to prove some stuff like we want to construct free monoids or other similar stuff for that we have several notions of natural number object that all coincide assuming the existence of such an object we can prove that we have free monoids for example but if you don't assume it we cannot so it's it's independent of the axioms of the elementary topos all right so this was in something like a because of you so depending on who you are just might or might not have been very familiar to you so now the next step is I want to move into the world of like phoenix category stuff and once you okay how does this how does this picture look like in phoenix categories do we have do we have natural number objects or not and what do they imply and because I'm now using the the word in phoenix category then I should like say something about what I mean by that uh and so let me just give you like a like a one slide summary so if if you're already familiar or if you did attend the first lecture of the the Hayatopos school last week by by Charles Resk then you should probably be already familiar with the phoenix categories some extent if not then just take this one I will just give this one slide summary so for the purpose of this talk and in phoenix categories just like there's a collection of objects or x y whatever and for any two objects there's a mapping space and that is really important I really want to fix that I have a space of maps from x to y where space can either mean topological space or con complex that doesn't matter and then that those spaces should give me some sort of weak enrichment meaning that whenever I have like two maps so that the codomain domain are appropriate I have some kind of composition but the composition only chosen up to some contractable ambiguity and then that composition is also associative but again associated only after some homotopies and and so on and all the classical categorical terms all the classical definitions of category theory limits adjunctions, Cartesian closure, adjoint functor theorem, presentability all all of those notions appropriately generalized to this infini category setting and that can be done either using like quasi categories or an infinity cosmoi or complete single spaces but for the purpose of the talk this is all I need like I will use some terms like high category terms but just believe that they exist in a primary category so now but just one slide we all have mastered infini category theory so we can move on to what infini categories you want to focus on so the main goal is to apply this to an elementary infini intervals and as the title suggests and so how you want to define elementary infini intervals we will certainly want the conditions that you can already find in an elementary intervals i.e. finite limits and co-limits, local collision close and sub-object classifier and then you additionally want some kind of object classifier universe the the issue is the exact notion of like universe you want is a little bit up to debate so there's like you can kind of decide on how strict the universe should be or how functorial how close it is on a certain construction and the claim I want to make is that it really doesn't matter for the purposes of this talk whichever notion of universe you think of the result I claim holds so in order to kind of make this stronger claim I would just not use the universe and we use some weaker condition that just follows from any notion of universe that is out so instead of using universe I just use finite descent and a finite descent is is like a is like the descent condition which was introduced by also by Charles Reckon in the second lecture and it's about like the interaction between pullbacks and pushouts the finite I'm using here is only because well we don't have infinite co-limits like the definition here is finite co-limits and so for finite co-limits we have the descent and it's the same condition follows from any notion of universe that is out there and that also suffices for the result I want to state so I don't even need to go to the strength of the whole universe I just need the descent condition that's enough and before I move on let me note that as soon as you have an infinity category which has these properties I can take the subcategory of zero truncated objects and that subcategory will be elementary topos because it will still have finite limits, peak localization closed and have a subcategory as I can make a nice bonus and notice this definition does have like lots of examples the most difficult example is the infinity of spaces but then you also get pre-sheeps you also get all higher toposes in the sense of like lyrical risk or something but in addition to those you also get like not presentable examples such as filter product between toposes so there's like a bunch of examples of this condition I stated okay so now we have a certain infinity category we want to study what do we want to study about it natural number objects so what is an appropriate notion of a natural number object in the infinity setting and because I've already defined them in the one setting I'm just like a bullet summary how they would look like in the infinity setting and for freight and piano the generalization is pretty straightforward right so the freight natural number object is just you have two certain diagrams of co-limit diagrams and I can just make exactly the state statement I still have exactly the same diagram and just say put those on a co-limit diagrams but inside some infinity category and for a piano it's the same I mean I just want a certain sub-object that is closed under os to be the same as the original object again you can just phrase that and the fact that you know it's a kind of high category setting has no influence on on the definition. Lavian natural number objects are not like that so for Lavian natural number object now I need to actually strengthen the condition I need to say that whenever I have a triple of an object x and some endomorphism u with a map on the point b then I have a space of maps from the natural number object to x and I want that space to be contractable so that is very different from the more classical categorical setting where we have a set of morphisms and we want that to be unique so there's no uniqueness anymore there's only contractability so it's like a much weaker form um but yeah so piano and freight are pretty much similar I would say so we are defined our infinity categories and the natural number object so I'm finding a position where I can state the the main result and the main result is that if e satisfies the condition I stated before so it's looking like a region closed finality by complete and satisfies the standard has a salvage classifier then all three notions of natural object coincide and exist and so that last point is very important but unlike the classical setting I can prove the existence from these accidents and also that they coincide okay so let me give like a summary of the proof so in an ideal world I would like to spend the next like I guess 25 minutes on this proof but I don't have 25 minutes so that means I have the next five minutes on this proof and the fun fact about the proof is that it breaks down nicely into like three almost completely independent steps and so the first step is motivated by results from I guess any introduction to algebraic topology course and in school right so so if you go like any intro to algebraic topology course the first result they will probably teach you is that pi one of s one is that and so I want to categorify that result I'm going to take a more general view of that and so how can I do that I just think of okay that is the free group on one generator so really what I'm saying is that pi one of s one is the free group on one generator and then even more generally okay if I don't want really pi one of s one I just want the loop loops on s one so pi one is like the the homotopy class of loops but I can just ignore the homotopy part and just say I want loops on s one and then loops of s one can be expressed in any category with finite limits and columns so that is not familiar then you just have to test beyond that that I can always write a circle as a certain column and I can write the loops on the circle as a certain limit as a certain pullback and now the statement then becomes if you have a category with like these settings as a matter of fact all I need is finite limits finite co-limits and descent then I can form loops on s one and loops on s one will be the free group on one generator it will satisfy the universal property here and so that's like the first step and it's motivated clearly by like background knowledge and algebraic topology and then the next step is if you recall as soon as I have this e I have an underlying elementary topos and now you just use kind of very classic literature on this that proves that it's whenever you have elementary topos and you have the free group on one generator you have a natural number on you can just construct it right so you use this object classifier and you carve out the n out of the z basically so that's like a very kind of summary step of the pool right that when you have z and it's like sub-element like a sub-object of that and just get it out of and so after step two then you prove that the underlying elementary topos has a natural number object what you want then to prove is that okay this underlying elementary to this national object in the underlying elementary topos is also not an object in the whole infinity category so there's like a lifting kind of argument and that doesn't follow obviously at first the thing is as I explained so the freight and piano conditions are very similar so as soon as they're holding the underlying one category they will hold also in the infinity category there's not that much more effort like you have a certain column of diagram and you just want to check that the same column of diagram is also a column of diagram in the infinity category so there's there's some work but it's not too bad the the main challenge of the proof and something like the hardest step is to show that this if you assume that's a love your natural number object in the underlying elementary topos then it is also love your natural number object in the infinity category and that is quite challenging because now you're making it like you're strengthening the claim quite a lot you're going from a statement about uniqueness of maps to contractability of some mapping spaces and that's just not obvious and that is also the part where unfortunately you cannot just use elementary topos methods because elementary topos don't really work well with like homotopy stuff and so the trick then is is to like use the wisdom of homotopy technologists because they they have been digging a lot about how logic and homotopy theory cannot be tracked so it's not actually homotopy type of per state but just look at the work and like try to get some ideas from there and adapt it to like the categorical setting and using that you can then find an appropriate proof that if you have a love your natural object in the underlying elementary topos it will also be a love your natural object for the full thing all right cool i still have a lot of time so i want to move on to the who cares so now that you have a natural object we can use it to do what's the point of it and let me focus on like three examples and one future direction so the first fun fact is so let me give you some history um the the good nick topos is it can be lifted to a higher topos like an infinity topos and it is in fact that whenever you pick any good nick topos you can find an infinity topos which has that good nick topos as a zero truncation and that infinity topos is often called the enveloping infinity topos so this was already known 10 15 years ago and one thing one might have expected is that something similar holds in the elementary setting so whenever i have an elementary topos i can lift it to some elementary infinity topos in a similar way but this result implies that this cannot be true it's just impossible right so for example the category of finite sets because it doesn't have a natural number object can never be the zero truncated objects of any elementary infinity just not going to happen and in particular this also implies that the category of finite spaces cannot be elementary infinity topos because it doesn't have natural number object and so if you want some kind of like finitary version of spaces that has that is still like a topos you have to like aim bigger so you need to focus on like couple small spaces for some couple that's like satisfy some properties and this has been studied by Loma and Aqua and make a recent paper and like this is like a like a nice contrast and i don't know it is i found it surprising and another like fun example and fun implication of the existence of NNOs is that we can like study um infinite columnets and so there's two aspects that we can use NNOs to study them one is that okay when we have a natural number object we can kind of reduce and determine quite straightforwardly when our infinity category does have infinite columnets and kind of reduces down to having a couple of the terminal objects so it's like a useful fact on studying infinite columnets in this elementary infinity topos setting i think more importantly it allows us to like define so-called internal sequential limits and like unfortunately i don't have enough time to like break down what they are but we can define like internal sequences inside our infinity categories and then define like the internal sequential limit of that internal sequence as a certain co-equalizer which then exists that's nice and like using the internal sequential limit you can actually construct negative one truncations um this is again one of those results that in the ingredient infinity holds by default and that was also part of the lectures of the summer school i think that lecture number three um but and there you just use like local presentability to use it but here you don't have local presentability so you use like NNOs to kind of circumvent that and get around decision and this should remind us of like a similar result in the elementary setting of epi-mono factorization right so one of the classical results of elementary topos is that you can always factor every map into an epi followed by mono and so this is a straightforward generalization of that like negative one truncation is precisely like epi-mono factorization of the map so we cannot just do it without the NNO but with an NNO we can get the same result um as a last step so one key example of the application of the natural number object in the elementary topo setting was the existence of free monoids so one natural question that would be okay you have to do the existence of a natural number object so can you now use it to do the existence of free like monoids in your infinity category setting and the answer is kind of a little disappointing so i i guess the answer is i don't know i don't know yet at least and not only that i don't know but it turns out this question is kind of quite challenging and so let me end this talk by giving you a sense of what the challenge here is when you're trying to like approach this question um if unlike the the one category setting in order to define monoids in some kind of infinity category you use you need to use some kind of like like algebra gadget like so operas or the view theories or something that is because you need to have operations for all possible levels but then ironically it can operate itself the definition of an upright and sometimes depends on the natural numbers so you have an operation for every n and so now the natural question you can ask yourself is should you use the natural number object that exists inside the topos or should you use the external one that exists in spaces and now you might want to okay what's the difference but it turns out there is a difference like in in certain elementary toposes there are so-called non-standard natural number objects which have additional uh non-standard natural numbers and so the definition would be different like you would get a new and different definition of an upright which has like additional operations and yeah that would just not coincide today the first question that could you ask here is when you say monoid what do you mean like do you mean the monoid where you have just the standard operations or where you also have the additional ones and so like not only like is it challenging to construct it it's even challenging to define what the appropriate notion should be and this challenge doesn't arise in the in like in the one setting right so when you try to define a monoid in certain elementary topos you never face the challenge regardless of what the natural number object is you never face the challenge of having to define additional operations because it always suffices to like define operations for level like zero one and two you never need the higher levels because your associativity everything will be strict this is kind of the challenge that only arises when you try to study um homotopical algebra like so higher algebra setting so there seems to be like this um surprising interaction between non-standard natural number odds and higher algebra that is clearly not well understood and definitely merits for the study so with this um I end my talk thank you very much