# Examples of toposes for analysts

I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.

Can you provide some examples of toposes which are of use to analysts? e.g., toposes where the objects consist of topological spaces, measure spaces, etc.

• Not to take away anything from topos theory, but I think it's common practice for near every under-represented theory to portray itself as extremely important to modern mathematics. It's probably more neutral to just say certain specific people believe subject X is important to modern mathematics, and make citations. This could eliminate a lot of confusion among readers. – Ryan Budney Dec 18 '13 at 16:20
• @Ryan, do you have any examples of toposes which might be useful to analysts, or any reason why toposes are of no interest to analysts? If not, I don't see what your comment has to do with my question, since I don't see any comments indicating confusion among readers. – Tom LaGatta Dec 18 '13 at 19:05
• The purpose was to help give readers some context. I suppose if the answers were only to be read by you my comment would be irrelevant. – Ryan Budney Dec 18 '13 at 19:42
• I wonder if there is any analyst who said, "Toposes are extremely important in modern mathematics." – Gerald Edgar Dec 19 '13 at 15:21

First realize that the right definition of orbifold is build on topos (the definition without topoi are mostly cheating).

Here is an example where similar things appear as limit of Riemannian manifold. [For more info, check "On the long-time behavior of type-III Ricci flow solutions." by Lott or the appendix in paper "Diffeomorphism Finiteness, Positive Pinching, and Second Homotopy" with Tuschmann.]

Consider the circles $\mathbb S^1_\ell$ of length $\ell>0$. As $\ell\to 0$, in Gromov--Hausdorff topology they converge to point. On the other hand, the space of harmonic 1-forms stay the same on $\mathbb S^1_\ell$, so for some problems Gromov--Hausdorff topology is not right choice.

You may think of $\mathbb S^1_\ell$ as $\mathbb R/(\ell{\cdot}\mathbb Z)$. Then it is natural to think that $\mathbb R/\mathbb R$ is the limit of $\mathbb S^1_\ell$; More precisely $\mathbb R/\mathbb R$ has one chart $\mathbb R$ and all the shifts $x\mapsto x+a$ as the transition maps. So the limit is smooth, but not a manifold. You may think of it as a pseudogroup of transformations, but you need topoi to give an invariant definition (the same story as with orbifolds).

• Could you develop or give a reference on the fact that "the right definition of orbifold is build on topos" ? Because apparently I've never seen this definition... Does it mean that an orbifold can be defined as some kind of structure on an underlying separated topos ? thank you ! – Simon Henry Dec 19 '13 at 11:02
• I think he just means that orbifolds are most naturally thought of as stacks, i.e. sheaves of groupoids on a category with a Grothendieck topology, and sheaves on a category with a Grothendieck topology is an (or the) example of a topos. – Dan Petersen Dec 19 '13 at 12:25
• Maybe, but after some research, It seems that the paper "Orbifolds, Sheaves and Groupoids" Of Moerdjik sugests that orbifolds can also be defined as separated toposes with a structural sheaf which are locally manifolds (in the sense that they admit a slice which is a manifolds) – Simon Henry Dec 19 '13 at 14:01

There are many examples.

You might like Rousseau's reduction of many complex variables to a single one.

In general, topos theory allows one to avoid the axiom of choice and classical logic. For instance, see our concrete treatment of parts of the theory of Banach algebras.

These ideas of course have their roots in the work of Banaschewski and Mulvey of finding a proper treatment of sheaves of C*-algebras/ C*-bundles by considering a single C*-algebra in a sheaf topos. This work is also used in the Bohr toposes mentioned above.

• Rousseau's paper looks interesting. Might be a bit of an overstatement to say it is a "reduction": he reproves a rather formal structre theorem. I would be interested to see a more "hard analysis" theorem rephrased in the topos language: for example, can he handle the Levi problem? – Steven Gubkin Dec 19 '13 at 16:26

Terence Tao's cheap nonstandard analysis can be interpreted as taking place in a topos related to the topos $\text{Set}^{\mathbb{N}}$ of sets indexed by the natural numbers; see this math.SE question for details.

You should Google for "synthetic differential geometry".

• By the way -- I have seen this answer in the "Low Quality Posts" review queue. Maybe just because it's rather short ... . – Stefan Kohl Dec 18 '13 at 17:57
• Stefan Kohl: That surprises me, but I suppose that if this were really a low quality answer, I'd be one of the last to recognize it. Barring some explanation, though, I stand by the answer. – Steven Landsburg Dec 18 '13 at 18:34
• @StevenLandburg, synthetic differential geometry is interesting, but I don't see how this answers my question regarding specific toposes of interest to analysts. The entire approach of synthetic differential geometry seems to begin synthetically rather than analytically: with a topos $E$ in hand, differental geometric structures are built. My question is in the opposite direction: what are some such toposes $E$ which might be of interest to analysts? ncatlab.org/nlab/show/synthetic+differential+geometry – Tom LaGatta Dec 18 '13 at 19:00
• @TomLaGatta: I'm not familiar with the literature on toposes and lexicographic preferences (and my initial uneducated reaction is to wonder how toposes could possibly be useful in this context), but you've inspired me to dig around a little. If I find something interesting, I'll report back. – Steven Landsburg Dec 18 '13 at 21:33
• @TomLaGatta, not exactly true. Synthetic diff. geom. is more about providing a model which allows one to work with infinitely small or infinitely large numbers. In this sense it is like a topos-theoretic version of nonstandard analysis, however it has categorical rather than logical flavor and allows a wider variety of models (e.g. existence of nilpotent infinitesimals and analogies with algebraic geometry). – Anton Fetisov Dec 19 '13 at 20:43

Besides the important examples of topoi already mentioned (like SDG), I would argue that the most important topos for any analyst is just the topos of sheaves on some topological space. I assume that the statement "sheaves are important for analysts" is well-known and doesn't require further argumentation. The study of topoi naturally focuses our attention not on model-dependent results, bizarre axioms and exceptional cases, but on general phenomena and parameter-dependent versions of statements, since the objects of a sheaf topos are naturally dependent on a point in the base space. For these reasons topos theory advocates constructive approach to mathematics, not because of some philosophical reasons, but because logical and set-theoretic intricacies simply don't make sense once you start working over a general base. For example, the axiom of choice fails unless our topological space is discrete. Another example: if you want to do some parametrized topology, you really should consider moving from topological spaces to locales, since general topological spaces over a base are very, very badly behaved.

From this point of view topos theory is some abstract machinery that allows you to transform a good enough (i.e. constructive) proof of any theorem into its parametrized version.

Things get even more fun once you become interested in some homotopical phenomena and deformation theory. It can become very tricky to prove something without considering $\infty$-topoi, either explicitly or shyly hiding them.

• Thank you for your excellent answer, @Anton. Could you expand more on the statement, "sheaves are important for analysts"? Coming from the world of probability, we don't talk about sheaves very often, and I would like to know more about why we should. – Tom LaGatta Dec 19 '13 at 21:45
• @TomLaGatta, since I myself am rather far from analysis, I can't be really specific here. I can only give some pointers. Firstly there is the work of Mikio Sato et al. on hyperfunctions, which are a special sort of generalized functions, but are defined and studied via sheaf theory. Secondly, most interesting analytic objects over manifolds (differential forms, tensor fields, connections etc) naturally form sheaves and that part of structure is very important. Any good algebraic complex geometry book will discuss it (e.g. de Rham's theorem). – Anton Fetisov Dec 19 '13 at 23:35
• Finally, see Kashiwara, Schapira "Sheaves on manifolds". It discusses a far-reaching generalization of previous themes, with applications to the study of differential equations on manifolds. It's definitely not the book to get a first acquaintance with sheaves, but it gives a nice view of (20 y.old) state of the art. – Anton Fetisov Dec 19 '13 at 23:38
• I don't really know how sheaves can help in probability theory, but my view of probability doesn't exceed a one-semester undergraduate course. I heard the word about something called "stochastic geometry" which looks like ag/prob mix, and I also assume that anyone can benefit from the study of PDE (e.g. for stochastic motion). If you also count statistical quantum physics as probability, then you should also be interested in things like quantum field theories, and these have a deep relation to sheaves etc, but that looks far too deep for any introduction. – Anton Fetisov Dec 19 '13 at 23:43

I don't know if it counts for "analysis", but toposes (=topoi) appear to be related to quantum mechanics. There's even a book on the subject. For a specific example, a keyword might be Bohr topos.

The point of talking about topoi is rarely to prove something new or interesting about a particular category of spaces. Quite the opposite, the point is typically to prove something interesting about topoi which applies to all categories of spaces that also happen to be topoi. The point is that the abstraction is powerful enough to prove quite a bit for many interesting categories.

Categories of Banach ($\textbf{Ban}$ of all and $\textbf{Ban}_{fd}$ of finite dimensional) and Hilbert spaces ($\textbf{Hilb}_{\oplus}$, $(\textbf{Hilb}_{\oplus})_{fd}$, $\textbf{Hilb}_{\otimes}$, and $(\textbf{Hilb}_{\otimes})_{fd}$) (and related spaces of interest to analysis) can be seen as symmetric monoidal categories, with additional structure. Like topoi, monoidal categories are important and powerful structures with many important theorems proven in them.

These categories are Galois adjoint to topoi in ways that allow natural internal logics to be built and find consonance with the adjunction. This is one way in which topoi are seen in relation to these symmetric closed monoidal categories. You see this association, for instance, in operationalist reductions of quantum theories (which are formulated over a category of Hilbert spaces).

But although these adjoints have some structure-preserving properties, I have found that many treatments stick with the natural categorial structure and prove things in the natural internal logics. Braided monoidal categories have a huge amount of structure and really, I think many will agree that two of the most important types of categories to study when learning categorial methods are topoi and monoidal categories, as that already gets a good part of modern mathematics.

However, if you are interested in topos theoretical direction exclusively, another direction important in analysis besides that above and those of other answers is Pestov's Conjecture, that there exists a Grothendieck topos whose internal Banach spaces are operator spaces. This direction of research is intended to provide a natural category that shows which theorems on Banach spaces extend to operator spaces in general.

• Is there a reference for Pestov's conjecture? A quick search doesn't seem to reveal any relevant sources. – Dmitri Pavlov May 31 '17 at 13:58