45 votes
Accepted

Grothendieck says: points are not mere points, but carry Galois group actions

Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) ...
Qiaochu Yuan's user avatar
36 votes
Accepted

Higher Topos Theory- what's the moral?

It seems there are really two questions here: Why higher category theory? What questions can you pose without the language of higher category theory which are best answered using higher category ...
Tim Campion's user avatar
  • 61.5k
34 votes
Accepted

Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

[Update 2024-04-15: The preprint The countable reals is now available.] Please allow me to list some basic observations that might clear up things. I work constructively (without excluded middle) and ...
Andrej Bauer's user avatar
29 votes

Higher Topos Theory- what's the moral?

I'm going to give a general answer first, and a specific answer below. It is my opinion that when Jacob Lurie wrote Higher Topos Theory, he was channeling Grothendieck. When Grothendieck ...
David White's user avatar
  • 29.8k
27 votes

Can the opposite of an elementary topos be an elementary topos?

The answer to the question in the title is no, assuming you want to exclude the trivial case of the terminal category. Let $E$ be an (elementary) topos whose opposite is also a topos. The initial ...
Tom Leinster's user avatar
  • 27.2k
27 votes

Interview of Connes, Caramello, and L. Lafforgue about topos theory

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than ...
25 votes

Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

In the topos we construct in our paper there is a surjection/epimorphism from the natural numbers to the Dedekind reals. In the model of CZF you mention (and in the effective topos) the Dedekind reals ...
James Hanson's user avatar
  • 10.3k
24 votes

Higher Topos Theory- what's the moral?

Here is a belated addendum to the other answers. In short, I think the comparison with Grothendieck is on point. More specifically I want to argue that HTT accomplished for higher category theory what ...
Dan Petersen's user avatar
  • 39.3k
23 votes

Two interpretations of implication in categorical logic?

There are two concepts here, which are tightly connected. Logically, this corresponds to the distinction between $\vdash$ and $\Rightarrow$. (A) Morphisms $t : \Gamma \to A$ represent (well-formed, ...
varkor's user avatar
  • 8,755
22 votes
Accepted

When does a topos satisfy the axiom of regularity?

The relationship between toposes and set theories was studied comprehensively in Steve Awodey, Carsten Butz, Alex Simpson, Thomas Streicher: Relating first-order set theories, toposes and categories ...
Andrej Bauer's user avatar
22 votes
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Is the opposite category of commutative von Neumann algebras a topos?

The opposite category of commutative von Neumann algebras is not a topos because categorical products with a fixed object do not always preserve small colimits. See Theorem 6.4 in Andre Kornell's ...
Dmitri Pavlov's user avatar
22 votes
Accepted

Are the models of infinitesimal analysis (philosophically) circular?

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of ...
Joel David Hamkins's user avatar
22 votes
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Condensed vs pyknotic vs consequential

Some comments: Regarding 1): They are quite different. Johnstone actually uses a very general notion of "cover" in his sequential topos -- his site is a full subcategory of metrizable ...
Peter Scholze's user avatar
20 votes
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Locales as geometric objects

First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works. Here are some ...
Simon Henry's user avatar
  • 40.2k
20 votes
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The philosophy behind local rings

I'm not sure if this constitutes a full answer, and a lot of it has already been said in some form by HeinrichD in the comments. Because your question is ultimately one of philosophy, I will focus ...
R. van Dobben de Bruyn's user avatar
20 votes
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Precise relationship between elementary and Grothendieck toposes?

There are known statements that are true in any Grothendieck topos, but not in every elementary topos with NNO. For instance: Freyd's theorem that a complete small category is a preorder is not ...
Mike Shulman's user avatar
  • 65.1k
20 votes
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Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology

Marc's examples are good ones, but let me add two more (which are closely related to each other): 1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let ...
Jacob Lurie's user avatar
  • 17.6k
20 votes
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Surmounting set-theoretical difficulties in algebraic geometry

Let me start by discussing a bit the option of having a large class of generators. You might be interested in the notion of locally class-presentable. To be precise here, I need to be a bit set-...
Ivan Di Liberti's user avatar
20 votes
Accepted

Why do elementary topoi have pullbacks?

I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. However, this is based on the definition of sub-...
Simon Henry's user avatar
  • 40.2k
20 votes

Resources for topos theory

For a beginner, the more accessible textbooks seem to be the following two. Francis Borceux, Handbook of Categorical Algebra, Volume 3. Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and ...
Dmitri Pavlov's user avatar
20 votes

Is there a good general definition of "sheaves with values in a category"?

In my view, the correct notion of "sheaf of Xs" is "internal X in the topos (or $\infty$-topos) of sheaves of sets (or spaces)". (I mentioned this previously on MO here.) Since ...
Mike Shulman's user avatar
  • 65.1k
19 votes
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"Spatial (geometrical)" realization of Elementary topos?

I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes. My main ...
Simon Henry's user avatar
  • 40.2k
19 votes

Recommendations to learn about the use of toposes in logic?

In no particular order: Mac Lane & Moerdijk, Sheaves in Geometry and Logic (1992). Clearly explains things like what a Grothendieck topology and a Lawvere-Tierney topology are, and gives some ...
Gro-Tsen's user avatar
  • 30.1k
19 votes
Accepted

What is known about the homotopy type of the classifier of subobjects of simplicial sets?

It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the ...
Peter LeFanu Lumsdaine's user avatar
18 votes
Accepted

(Co)complete topoi that are not Grothendieck?

A classical example is $G$-$Set$ for a large group $G$. That this is a cocomplete elementary topos is not hard to see. Limits and colimits are formed at the underlying set level, and exponentials $Y^X$...
Todd Trimble's user avatar
  • 52.4k
18 votes
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Toposes with only preorders of points

$(i) \Leftrightarrow (ii)$ is true and is Proposition C.2.4.14 in Peter Johnstone's Sketches of an elephant. More generally he shows that a bounded geometric morphism $f: \mathcal{E} \to \mathcal{S}$ ...
Simon Henry's user avatar
  • 40.2k
18 votes

What are the points (and generalized points) of the topos of condensed sets?

The category $\mathbf{Cond}$ of condensed sets is equivalent to the category of small sheaves over any of the following three large sites. (For small sheaves, see Mike Shulman's paper Exact ...
Alexander Campbell's user avatar
18 votes

What is neutral constructive mathematics

You'll probably have better luck with the phrase "intuitionistic higher-order logic" (IHOL). A good place to start is the book by Lambek and Scott, Introduction to Higher Order Categorical ...
Todd Trimble's user avatar
  • 52.4k
18 votes
Accepted

An extension of the Galois theory of Grothendieck

The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be ...
Simon Henry's user avatar
  • 40.2k
18 votes

Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi

There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of ...
D.-C. Cisinski's user avatar

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