37
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 ...
- 62.6k
35
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 ...
- 47.9k
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 ...
28
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 ...
- 27.4k
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 ...
- 10.4k
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 ...
- 39.5k
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, ...
- 9,521
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 ...
- 47.9k
22
votes
Accepted
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 ...
- 37.2k
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 ...
- 230k
22
votes
Accepted
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 ...
- 20.9k
20
votes
Accepted
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 ...
- 40.8k
20
votes
Accepted
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 ...
- 27.9k
20
votes
Accepted
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 ...
- 65.8k
20
votes
Accepted
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 ...
- 17.7k
20
votes
Accepted
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-...
- 8,475
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-...
- 40.8k
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 ...
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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 ...
- 65.8k
19
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 ...
- 13.4k
19
votes
Accepted
Properties of pyknotic sets
Let me recall a little bit of the background. The question is about the relation between topological spaces and pyknotic sets, and properties of the topos of pyknotic sets. Recall that pyknotic sets ...
- 20.9k
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 ...
- 30.8k
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 ...
- 19.6k
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 $Y^X$...
- 52.6k
18
votes
Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
I'm not an expert on the sheaf-theoretic approach to probability theory, but a quick look at the paper you're asking about shows that it's a 6 page conference proceeding from 2017 that defines a new ...
18
votes
Accepted
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 $f: \mathcal{E} \to \mathcal{S}$ ...
- 40.8k
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 ...
- 5,479
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 ...
- 52.6k
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 ...
- 40.8k
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