The video of my recent talk "Toposes as unifying spaces: historical aspects and prospects" at the *Workshop in honor of Alexander Grothendieck’s legacy* at the Universidad Nacional de Colombia is now available on YouTube:

This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.

It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series *Lectures Grothendieckiennes* organized by Frédéric Jaëck (whose slides are available here):

In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.

]]>Concerning 2, in saying that results could be obtained “without any creative effort” through the ‘bridge’ technique, I wanted to emphasize that, once an equivalence between two different presentations of the same topos is established, the calculation of how invariants express in terms of the two presentations is essentially canonical and can be automatized in many cases (descriptions of classes of invariants for which such calculations can be performed in a semi-automatic way are provided for instance in this paper and this other one). Of course, the complexity of such calculations can greatly vary depending on the given invariant (many topos-theoretic invariants can be readily calculated, while cohomological invariants tend to be very difficult to compute even in special cases); still, conceptually, such 'unravelings' are just computations, which, at least in principle - as well as in practice for many invariants! - do not require a creative effort to be performed. On the other hand, identifying the ‘right’ toposes, and the most appropriate invariants on them, which incarnate the “essence” of a certain mathematical situation or capture some specific feature of a given problem, is in general quite subtle and may require an imaginative effort. Here are a couple of relevant excerpts from my habilitation thesis, where these issues are thoroughly discussed:

*We have already hinted above to the fact that there is an strong element of automatism implicit in the ‘bridge’ technique. In fact, in order to obtain insights on the Morita-equivalence under consideration, in many cases one can just readily apply to it general characterizations connecting properties of sites and topos-theoretic invariants. Still, the results generated in this way are in general non-trivial; in some cases they can be rather ‘weird’ according to the usual mathematical standards (although they might still be quite deep) but, with a careful choice of Morita equivalences and invariants, one can easily get interesting and natural mathematical results.*

*…*

*Starting from a topos or a Morita-equivalence, the calculation or expression in terms of sites or theories of presentation of topos-theoretic invariants is often technically non-trivial but feasible. On the contrary, trying to go in the other direction from a very sophisticated concrete mathematical result to a Morita-equivalence which could generate it is in general very difficult, if not impossible. In other, more metaphorical, words, this methodology generates a ‘rain’ of results falling in a territory surrounding a given problem whose essential aspects have been encoded by means of suitable topos-theoretic invariants. It is difficult to predict exactly where the single drops will fall, but, as the rain will eventually cover more and more of the wet space, so the application of this methodology is liable to bring a lot of concrete insights on aspects related to the original problem which could eventually lead to its solution. *

As to question n. 3, the technique of toposes as ‘bridges’ is based on the possibility of presenting a topos in multiple ways (for instance, by using sites, geometric theories, topological or localic groupoids, etc.) and of investigating topos-theoretic invariants from the points of view of such different presentations. Going from ordinary toposes to higher toposes, many classical topos-theoretic tools and techniques can be naturally extended (as shown in Lurie’s book “*Higher Topos Theory*”), but the theory of ‘presentations’ for higher toposes is more complex than its classical counterpart: for example, higher sites are no longer sufficient to capture all left exact localizations of a higher topos. Also, from a logical point of view, the correct analogue of geometric logic has not yet been identified (most research efforts have concentrated so far on identifying the higher analogue of the internal logic of a topos, rather than on generalizing geometric logic to the higher setting). I certainly expect the technique of toposes as ‘bridges’ to be fully applicable in the higher setting, once an effective presentation theory for higher toposes (using ‘computationally effective’ generalizations of higher sites, or higher geometric theories, or any other suitable mathematical objects susceptible of presenting higher toposes) will be developed.

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- This question quotes Laurent Lafforgue (without naming him for some reason) comparing Caramello's program to the Langland's program. Their specific question is: "I wonder if there were any papers confirming the overall sentiment of , i.e. papers where Caramello's theory is applied to Langlands. Were there at least some long-standing problems that historically were not believed to have any serious connection with logic or topoi, and were solved using Caramello's techniques?"

Ignoring their dubious claim that, "The entire mathematical world has produced few, if any, correspondences which unify lots of non-trivial mathematics as elegantly as the Langlands program does," it would be nice to know what the current state of progress of linking these two programs is. - This question, by far the highest profile, is asking for insight into the quote "one can generate a huge number of new results in any mathematical field without any creative effort," and whether it is an exaggeration.

The answers are divided based on their interpretation of the quote, but Caramello's book (published since the question was asked back in 2010) certainly contains numerous illustrations of the kinds of result that can be extracted from the topos-theoretic bridge framework. As someone who works with Grothendieck toposes myself under Caramello's supervision, I know that "without any creative effort" doesn't mean "without any effort": extracting the results in a form that is comprehensible takes significant work, but the reason that effort is not 'creative' is because the toposes act as a guide, avoiding the need for one to rely on great leaps of inspiration to progress. I hope Prof Caramello will have more to say on the subject. - This question asks how Olivia's program relates to Jacob Lurie's work on higher topos theory and higher categories, specifically asking, "Has there been any work connecting these schools of thought?"

Most of the results in Lurie's*Higher Topos Theory*are extensions to oo-toposes of results for Grothendieck (1-)toposes, so there is some immediate relation between the background on which the programs rest, but it would be interesting to get some insight into how Prof Caramello views the relation between their research directions.

I may eventually report any interesting answers and discussion provided here as answers to the original questions.

]]>Thanks again to John Alexander Cruz Morales for organizing this course, and for making the recordings available with the kind help of Javier Gutiérrez.

]]>Thanks again to my excellent interviewer - mathematician Francesco Genovese - and to all the staff of Meet Science for the organization!

]]>Two of my joint publications with Jens Hemelaer have been published in the past week.

The first, *An Essential, Hyperconnected, Local Geometric Morphism that is not Locally Connected* is a counterexample to an open problem about geometric morphisms. Recall that the basic data of a geometric morphism f: \mathcal{E} \to \mathcal{F} consists of an adjoint pair of functors \left( f^*: \mathcal{F} \to \mathcal{E} \dashv f_*: \mathcal{E} \to \mathcal{F} \right) , the inverse image and direct image functors respectively, where f^* is required to preserve finite limits, but a geometric morphism can have many extra properties, including the existence of further adjoints for the constituent functors.

There are a handful of surprising coincidences amongst properties of geometric morphisms. A geometric morphism is called *locally connected* if its inverse image preserves exponential objects and the same is true for all slices of that geometric morphism (recall that for any object E of \mathcal{E} , the category \mathcal{E}/E is a topos, and it turns out that for f as above, we get a well-defined geometric morphism f/E : \mathcal{F}/f^*(E) \to \mathcal{E}/E whose inverse image functor sends \alpha: X \to E to f^*\alpha: f^*(X) \to f^*(E) ). A locally connected morphism automatically has a left adjoint f_! to f^* . This means that a locally connected morphism is a special case of an *essential* geometric morphism.

Recall that a geometric morphism is called *connected* if its inverse image functor is full and faithful, so that the codomain topos is equivalent to a full subcategory of the domain topos. In Johnstone's 2011 paper, *Remarks on Punctual Local Connectedness*, it is observed that for a connected and locally connected geometric morphism (or indeed a connected essential geometric morphism), there is a canonical natural transformation \theta: f_* \Rightarrow f_! obtained by applying full faithfulness to the composite of the counit of (f^* \dashv f_!) and the unit of (f_! \dashv f^*) . There are lots of surprising results in that paper, but the main theorem is as follows. Call a geometric morphism f *punctually locally connected *if it is connected and locally connected and satisfies the innocent-sounding extra condition that \theta is epic. Then f is automatically *strongly connected* ( f_! preserves finite products), *hyperconnected* (in addition to f^* being full and faithful, the image of f^* is closed under subobjects) and *local* ( f_* has an extra right adjoint f^! , which is automatically full and faithful). For bounded geometric morphisms, Johnstone also proves that the converse is true: a geometric morphism is punctually locally connected if and only if it is locally connected, hyperconnected and local.

To make sense of this powerful result, questions arise about how interconnected these properties are more generally. For example, we could have started with f being merely essential and connected; does that make a difference? If we impose hyperconnectedness and localness, does "locally connected" come for free, or is it actually independent? That specific question is what Jens and I provided a counterexample to in our paper, as an essential geometric morphism generated by a monoid homomorphism.

The second, *Solution to a problem of FitzGerald*, is more specifically about monoids.

The most obvious source of monoids is the collection of endomorphism monoids of objects in a category. Just as understanding the automorphisms of an object can be useful, the endomorphism monoid of an object is a nice invariant to have at one's disposal. Moreover, when a category has some structure, the monoids also inherit that structure: for example, endomorphism monoids in an additive (eg abelian) category acquire an additive structure making them into rings.

As such, it would be nice to know what we can deduce about endomorphism monoids based on the properties of their ambient categories. For example, Fitzgerald observed that when a monoid of endomorphisms of an object A in a category has commuting idempotents, then there are certain properties of the retracts and coretracts of A which can be deduced from that. He conjectured that in categories of algebras (categories admitting a monadic functor to Set), this becomes an equivalence: an object satisfies the retract/coretract properties if and only if its endomorphism monoid has commuting idempotents.

That's where Jens and I come in: we spotted that one such category of algebras is the category of actions of a monoid M, and that that category contains a special object A consisting of the monoid acting on itself. It is easy to show that the endomorphism monoid of A is exactly the monoid M. So in order to produce a counterexample to Fitzgerald's claim, we just needed to construct a monoid for which the equivalence fails.

More interestingly, we showed that any counterexample to the problem (in an arbitrary category) produces such a monoid, but that the converse *might* be false. Let A be an object in a category of algebras. For three of Fitzgerald's properties, if the monoid of endomorphisms of A has that property, then so must A, but the final property does not lift to A.

Finally, we invoke some general embedding results to show that there is some ring for which the conjecture fails, and we investigate some examples, showing that in some categories, Fitzgerald's conjecture does actually hold.

]]>What would be an interesting question is, given a topological group *G* acting continuously on spaces *X* and *Y*, satisfying my condition 2.1 in the notes (name suggestions are also welcome), with p \colon Y \to X an étale map, whether or not p_{/G} \colon Y/G \to X/G is also étale. I have a feeling it shouldn't be, but haven't been able to come up with a counter-example yet either.

All comments and corrections are very welcome.

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2/ What a good idea to see my collection of glyphs as a category ! Thanks for that.

But I go step by step, and before going in that way, I'ld like to check my approach on concrete and physical topics, in order to verify if my approach is consistant.

Just one example regarding relativity. In this article, I came to that point : there is a difference, in the way we refer to time and to space, roughly :

- For the space, the auteur
**𓂀**can discriminate between two positions of the Subject**𓁝**_{⊥}**𓁜** - For time, the Subject must be in
**𓁜**'s posture, so the relativity is between**𓁜**and**𓂀**.

My feeling is that difference in the postures leads to the difference between our representation of time and space (i.e. : between orthononality (real_{⊥}maginary) parts of an hamiltonian, and orthogonality between the 3 imaginary dimensions)... It is not yet finished, I have to progress on that point.

Another point : I find a general way of thinking that we can see in various fields in mathematics (see "ordre et chaos".

In this general overview, I know that I have to learn about jauge theory and obviously norm theory.

After that, and after making sure the approach is consistance and of some interest, I'ld like to follow your way : to see this set of glyphs as a category seems a great idea !

3/ That is a philosophical subject. I can just express here my own point of view : the Subject doesn't "exist" as the final object, but refer to the vacuum just as the initial one. That is more or less an Bouddhist approach... And in some way that of lacan and a lot of philosophers, for whom the Subject is definied as a being of speach.

4/ Yes I know that there is others categories, but I have to start by the begining. I define the "imaginary level" as a degenerative one, out of any other concept except the idea of "something" disturbes me. A cat that removes its paw by pricking on a heir, hurts itself to the so called "Real", and the entry in its imaginary passes throught that gate .

In the same way the path ↑ is just when the cat reconizes the disturbance, in reference to its passed experience.

After that, in , the human can imagine all the discrete concepts he can, as repetition, frequence, sucessor, time, order etc. It can also represent the arrow itself ↑, and go to Graph category and have other discimiante objects than {0;1}...

Using de Saussure's linguistic vocabulary :

- the first arrow ↑ is a "diachronic" concept,
- the reference ex post to that one, at level is a "synchronic" concept, just like any "object".

Regarding the paper itself : as its title recalls, it is an "abstract" of a presentation made on Zoom... Not at all an academic paper. And, as I have not finished to check this approach, it is not time to present an academic paper, just time to check, discuss, rewrite and correct it...

5/ The similarity lies in the posture of the Subject **𓁝 **which is the same in both cases.

Initial object is vacuum.

]]>the interpretation "Topos of -Sets" of

TTintoZFC

by: 'the interpretation *"Subtopos of -Objects"* of **TT** into **TTNNO**', and:

its composite set-theoretic semantics as the "Topos of -Sets equipped with a difference Y"

by: 'its composite interpretation as the "*Subtopos of -Objects, but equipped with a difference Y Object"* '.

And besides already being a good idea as most of generalizations that work, that is also one as it makes even clearer the fact that the actual duality here is between the *'difference'* and the *'-'* aspects, sets having no role in here. Thanks :)

2/ If you're committed to employing category theory, it's time to get you thinking more categorically. Here is the first example: as I suspected, the morphisms you draw are supposed to carry more structure than mere arrows. But the category of these five symbols and the transitions between them is too simple to "see" the richer structure. There are two ways to get around this. One is that you could make this into a richer category. As an ordinary category, you could add more objects and/or morphisms to "witness" the extra structure carried by the morphisms you already have; or, you could look to *enriched* category theory, where the 'collection' of morphisms between a pair of objects carries structure encoded in another category. An alternative (which essentially amounts to the first option from a different point of view), is that you could equip your symbols with more structure, identifying them as mathematical objects in some category which is rich enough to carry representations of the distinctions you conceptualise between these states. Often vector spaces are a good place to start with this kind of modelling.

The local/global example you give doesn't quite support your point, since curvature is a local property (the worm can measure the curvature by drawing big triangles, without ever leaving the surface) but I get some idea of what you are trying to say. But I still don't have a good understanding what the difference between your "postures" is, or even what they really are besides perspectives which are contrasting in some sense. I also don't know what "repeating a gap" means.

The distinction between parts and collections of elements sounds a lot like the distinction between classical topology (endowing a global collection of elements with a distinguished collection of open parts), which results in the concept of topological space, and point-free or point-less topology, which takes the open parts as the primary data, resulting in locales. There's plenty of material around on these ideas, which I hope you will find useful.

In category theory, it's often said that the definitions are the most important part, so if you have to omit anything from your presentation, make sure that the definitions stay in! In any case, if you manage to come up with definitions, that's a great starting point for applying any kind of mathematical reasoning.

3/ If I don't know what "the Subject" refers to, it's hard for me to follow anything. Regarding presenting a subject, there's a vast difference between a talk in a seminar where there are very restrictive time constraints (such as the talk of Lafforgue's you referred to) and a comprehensive presentation of a subject. If you take any course or textbook in maths, while there might be some motivating examples, the first real content consists of precise definitions of the objects or problems which are the subject of the course. If you needed or wanted to, you could fill in all of the detail that Lafforgue omitted, using the relevant academic papers and resources. Moreover, he can include technical terms without definition as long as he knows that the definitions are commonly known enough that his intended audience of mathematicians will be able to recall the standard definitions of these terms. I cannot do the same for your writing, partly because you use language from a discipline I am not well-versed in, but mostly because you give no references and offer no more intuitive explanation of your own.

4/ I gather that Lawvere places rather too much emphasis on the role of the category of Sets in category theory. Initial and terminal objects are not concepts unique to the category of Sets; they are defined in terms of their universal properties, and there are many other categories having initial and/or terminal objects. Crucially, their universal properties are *dual* to one another. That means that if I take a category \mathcal{C} with an initial object, and I consider its *opposite category* \mathcal{C}^{\mathrm{op}} (the category with the same objects but where all morphisms are reversed), then that object is now a terminal object. In the plain language of category theory, we in fact can talk about initial and terminal objects in exactly the same way! It's only the structure of the category of sets itself, which is the structure of a topos (although for this we only need that it's a *pretopos*) that makes the initial and terminal object behave in ways which aren't as symmetric. Could you explain more precisely what these postures are/what they do/what the difference between them is?

5/ Okay, these associations sound like analogies between philosophical ideas and the behaviour of objects/morphisms in the category of sets. I am not in a position to appreciate these analogies, for lack of understanding of the philosophy literature you reference. It is not clear for me, for example, why "the Subject never reach the Symbolism stage in an ex post position", or what this has to do with the initial object (in the category of Sets). Maybe you can at least clarify what property of the initial object you are appealing to?

What "topology" are you referring to at the end here?

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