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.

]]>- 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.

]]>\mathbb{T}'' \leqslant \mathbb{T}' \iff \mathbb{T}'' \vDash \varphi \vdash_{\vec{x}} \psi, \ \forall \, \varphi \vdash_{\vec{x}} \psi \in \mathbb{T}' .

I recently came across this in Olivia's Theories, Sites and Toposes (TST) where it is shown that \mathfrak{Th}_\Sigma^{\mathbb{T}} is a complete Heyting algebra. I'm wondering what other properties we are able to deduce.

I've attached some notes giving a sufficient condition for the open subtopoi to be complemented in terms of their corresponding theories, namely the theories of the form \mathbb{T} \cup \{\, \top \vdash_{[]} \varphi\,\} are complemented in \mathfrak{Th}_\Sigma^{\mathbb{T}} if there exists a sentence \psi such that \mathbb{T} \vDash \top \vdash_{[]} \varphi \lor \psi, \ \varphi \land \psi \vdash_{[]} \bot (i.e. \varphi is complemented).

As always, let me know if there are any mistakes. I'd like to know what else we can say about \mathfrak{Th}_\Sigma^{\mathbb{T}} (and generally generate discussion). Of course, just reading more TST would be sound advice as I've yet to finish it.

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This thesis is as follows: 1 / There is no change in the way of thinking since man lives in a group. 2 / To prove it, it is necessary to define what are the “thought patterns” of any Subject. 3 / The theory of categories can serve as a universal language model making it possible to identify these different patterns of thinking. 4 / This makes it possible to differentiate 3 levels of reflection, leading to

- The logic and elementary topos of Lawvere,

- A "topological approach", initiated by Évariste Galois and leading to Grothendieck's topos,

- A 3rd level where equivalences or "bridges" are established. 5 / This general scheme would make it possible to consider a "unitary theory" in physics. I leave on my blog the traces of my own evolution in this program. I try as I can, to discuss this subject within the Categorical Logic group led by Anatole Khelif at Paris Diderot. Not being a mathematician at all, I try to find relays to improve my approach. For a short summary of this approach : http://www.entropologie.fr/2021/01/resume-hec.html]]>

Here's a fun example I've experienced this week. We can assemble the collection of toposes into a 2-category in a few different ways, but I want to consider the 2-category of all toposes with natural number objects and all geometric morphisms between them. There are some results about limits and colimits in this 2-category in Johnstone's *Sketches of an Elephant*, but we can also ask some more basic questions, like "which (1-)morphisms are faithful in this 2-category?"

It turns out to be easier to answer the 1-dual of this question, namely "which geometric morphisms are cofaithful in this category?" A 1-morphism g: \mathcal{F} \to \mathcal{E} is cofaithful if it induces a faithful functor \mathrm{Hom}(\mathcal{E},\mathcal{D}) \to Hom(\mathcal{F},\mathcal{D}) by composition for all objects \mathcal{D}. For geometric morphisms, since we work with natural transformations between inverse image functors, this corresponds to composition with the inverse image functor.

Why is that an easier question to answer? This brings us to the logical perspective. The category of geometric morphisms \mathrm{Hom}(\mathcal{E},\mathcal{D}) is equivalent to the category of models of any geometric theory classified by \mathcal{D}, and this equivalence in natural in \mathcal{E}. As it happens, one of the simplest geometric theories is the *theory of objects*, \mathbb{O}, whose models in a topos are precisely the objects of that topos. Over any topos \mathcal{S} with a natural number object, we can construct a classifying topos \mathcal{S} for the theory of objects. In particular, given g: \mathcal{F} \to \mathcal{E} as above, we can consider \mathcal{D} = \mathcal{E} and deduce that in order for g to be faithful as a 1-morphism, the inverse image functor of g must be a faithful functor, which is to same that g must be a *surjection.*

Shifting to the perspective of toposes as categories and examining the data of natural transformations, we can show, conversely, that the inverse image functor of g being faithful is sufficient to make g cofaithful as a 1-morphism in the 2-category of toposes. Hooray!

Exactly the same trick allows us to show that the cofully cofaithful morphisms (I'll let you work out the definition of those!) are precisely the connected geometric morphisms, whose inverse image functor is fully faithful. And how do we understand those? Using a geometric perspective on toposes! A localic geometric morphism is connected if and only if it corresponds to a connected internal locale in the codomain topos, hence the name "connected".

I'd love to hear what perspectives people personally find most useful or familiar on toposes. I haven't even mentioned the internal language perspective, the (related but subtly different) mathematical universe perspective, the groupoid perspective, the sites and sheaves perspective, the stacks perspective, the algebraic gadget perspective... You can see where Johnstone's allegory of the Elephant comes from!

]]>A final draft of my article on supercompactly generated toposes has gone up on the arXiv. Due to it's length, I intend to submit it to the new Expositions journal in TAC; in the mean time, any comments are most welcome. Here is a summary:

A supercompact object in a topos is one which is simple or irreducible in the sense that any covering of it (any jointly epimorphic family of morphisms with this object as their codomain) must contain an epimorphism. A compact object is defined similarly, except that every covering is instead required to contain a finite subcover. These concepts arise very naturally in the setting of my ongoing work with toposes of monoid actions, because these objects correspond to the principal (aka cyclic) and finitely generated monoid actions respectively.

While proving results in my ongoing work about properties of toposes of topological monoid actions, I realised that many of the results relied only on the fact that these toposes have *enough *supercompact objects: every monoid action is a union of its principal sub-actions. As the volume of results exceeded what I strictly needed for studying monoid actions, I decided to devote a paper to toposes with this property, which turn out to be more common than you might expect. It turns out that toposes with enough compact objects behave similarly enough that I was able to explore them in parallel. Writing this paper also gave me the opportunity to present explicitly some special cases of the results in Prof Caramello's paper from summer of last year, which has lots of details about the relationships between sites and (co)morphisms of sites and the corresponding toposes and geometric morphisms.

I found some pleasing duality results and a bunch of examples which I hope people will find interesting.

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