Questions tagged [topos-theory]
The topos-theory tag has no usage guidance.
470
questions
3
votes
1answer
123 views
Sheaves on sites given by a (regular) cd-structure
Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...
7
votes
1answer
199 views
When do two topoi have the same cohomology of constant sheaves
Recently, I have some questions for some generalizations from algebraic topology.
I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same ...
7
votes
0answers
159 views
How much is known about the consistency strength of toposes and topos-like categories?
It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
14
votes
1answer
543 views
Can the opposite of an elementary topos be an elementary topos?
This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose ...
2
votes
0answers
262 views
Is the category of quasi-coherent sheaves not a topos?
There are two parts to my question:
Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?
My reasoning is thus:
It ...
9
votes
2answers
247 views
Simplicial spaces internally to simplicial sets
I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual).
After finding out about some of the uses of simplicial ...
21
votes
3answers
1k views
Is there a good general definition of “sheaves with values in a category”?
Let $\mathcal{A}$ be a category.
There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
5
votes
1answer
180 views
Hakim's definition of a locally ringed topos
In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:
(i) For ...
3
votes
0answers
44 views
Compact “subsets” in the internal real numbers of a topos
Let $\mathcal{E}$ be a topos with internal (Dedekind?) real numbers $\mathbb{R}_{\mathcal{E}}$. Suppose that $K\subset\mathbb{R}_{\mathcal{E}}$ is a compact "subset" (here, I assume that ...
10
votes
0answers
650 views
The “unification” of geometry via topos theory?
This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question.
There has been quite a lot ...
5
votes
1answer
266 views
Is there a Grothendieck correspondence for sheaves/stacks?
Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories
$$
\mathsf{DFib}(\mathcal{C})
\cong
\mathsf{PSh}(\mathcal{C}),
$$
whereas the Grothendieck ...
4
votes
0answers
71 views
Meta property criterion on the internal language of a topos of sheaves for Noetherianity
Let $X$ be a topological space. Then $X$ is irreducible if and only if in the internal language of $\mathrm{Sh}(X)$, $\bot$ is not true and we have one side of De Morgan's law, that is, for all ...
7
votes
0answers
128 views
Big Zariski topos and classifying topos of local rings
In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are ...
6
votes
2answers
264 views
Characterization of 'canonical' natural numbers objects
Canonicity is a useful property satisfied by some type theories, saying that every element of natural number type is propositional equal to an element of the form $s^n(0)$, where $s$ is the successor ...
12
votes
1answer
608 views
So after all, what is this thing about topos theory and non-binary truth?
Disclaimer. The question below is necessarily vague. I understand neither the subject matter topos theory nor the object about which my question is (the construction of a fractional / non-binary / ...
8
votes
1answer
254 views
A weak form of countable choice
Let $\Omega$ be the set/type of truth values. We're using constructive logic. Define
$AC_{0, 0} = \forall P : \mathbb{N}^2 \to \Omega, (\forall n \in \mathbb{N}, \exists m \in \mathbb{N}, P(n, m)) \to ...
13
votes
0answers
214 views
Which topoi are local with respect to Stone-Cech compactification?
Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
3
votes
1answer
261 views
Do stalks see epimorphism of stacks?
Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to https://mathoverflow.net/q/307366, this is an epimorphism if and only if it is almost surjective,...
23
votes
1answer
656 views
Is the opposite category of commutative von Neumann algebras a topos?
By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict ...
4
votes
1answer
223 views
Is $Set$ a tiny topos?
Let $Topos$ be the $(2,1)$-category of Grothendieck toposes and geometric morphisms. This is a $V$-sized, locally $V$-sized, locally locally small $(2,1)$-category with all small (2,1)-colimits (=...
11
votes
0answers
242 views
Describing the points of a glued topos
Let $f : \mathbf{X}\to \mathbf{Y}$ be a morphism of topoi; in his 1977 monograph, Johnstone describes the open mapping cylinder of $f$ as the following pushout of topoi:
$\require{AMScd}$
\begin{CD}
\...
5
votes
1answer
382 views
About an argument in Olsson's book
The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson.
I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end.
It seems that there might ...
12
votes
1answer
243 views
Stone duality for the algebra of Boolean functions such that $f(\top,\dots,\top) = \top$, or: What does the presheaf topos on $FinSet_\ast$ classify?
$\newcommand\FinSet{\mathit{FinSet}}\newcommand\FinBool{\mathit{FinBool}}\newcommand\FreeFinBool{\mathit{FreeFinBool}}\newcommand\Set{\mathit{Set}}\newcommand\Psh{\mathit{Psh}}$It's well-known that ...
12
votes
0answers
449 views
Is there a concrete application of topos theory?
The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But ...
17
votes
1answer
338 views
In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?
For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
3
votes
0answers
92 views
Is the category of covering spaces always a topos?
It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...
10
votes
3answers
699 views
Resources for topos theory
I am trying to learn topos theory and I am finding a strong scarcity of resources. Is there any canonical textbook to refer someone to when learning this topic?
So far, I have only been able to find ...
6
votes
1answer
389 views
Different definitions of condensed sets
The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the ...
8
votes
1answer
277 views
Is ${\bf Set}$ the terminal autological topos
An autological topos is a type of topos defined by Mike Shulman in his paper on stack semantics; specifically, they are toposes satisfying an additional topos theoretic axiom schema expressed in their ...
10
votes
1answer
928 views
Computations in condensed mathematics, page 32-34
I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:
At the last line of pg 32 - it seems to imply that ...
10
votes
0answers
345 views
How do properties of a partial order $\mathbb{P}$ affect the logic of the functor category $\mathsf{Set}^\mathbb{P}$?
$\DeclareMathOperator\true{\mathsf{true}}$I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway.
I am ...
13
votes
4answers
1k views
Two interpretations of implication in categorical logic?
I am a bit confused about the interpretation of "implication" in the standard treatment of categorical logic, for example in [Bart Jacobs 1999] "Categorical Logic and Type Theory".
...
10
votes
1answer
566 views
When does a topos satisfy the axiom of regularity?
In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-...
9
votes
1answer
313 views
From Topoi to Grothendieck categories
This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
8
votes
1answer
268 views
What's the localic reflection of a presheaf topos?
$\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.
What's the localic reflection of $\Psh ...
16
votes
1answer
470 views
Toposes with only preorders of points
For a Grothendieck topos $\mathcal{E}$, are the following assertions equivalent?
$(i)$ $\mathcal{E}$ is localic.
$(ii)$ The diagonal geometric morphism $\mathcal{E} \to \mathcal{E} \times \mathcal{E}$ ...
8
votes
1answer
171 views
Object classifiers in 1-toposes
In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to ...
13
votes
1answer
430 views
Does every category with a subobject classifier embed into a topos?
I've never seen an example of a category with a subobject classifier which didn't embed nicely into a topos. Is there a good reason for this?
Question 1: Let $\mathcal C$ be a category with a ...
3
votes
1answer
172 views
Alternative definition of power object in a category
The standard definition of a power object seems to be: objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \...
3
votes
1answer
183 views
Deligne's theorem for $n$-topos
Deligne's theorem states that a coherent topos has enough points, i.e. that we can prove that a morphism of sheaves on a "nice" site is an isomorphism by showing that the induced morphism on ...
10
votes
0answers
158 views
Subobject classifiers with a quantale structure
Quantales were kind of popular a few decades ago, as some sort of "quantum" version of the much more widespread concept of locale (Mulvey introduced them to study $C^*$ algebras). I was ...
8
votes
0answers
319 views
Geometric stacks, groupoids and étendues
If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent:
Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...
4
votes
1answer
289 views
Objects and morphisms in inverse limits of toposes?
Certain Galois toposes can be written as $\lim_{i \in I} \mathbf{PSh}(G_i)$ where $(G_i)_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of ...
7
votes
1answer
103 views
Prove that a Boolean two-valued topos in which supports split is well-pointed
In Lawvere and Rosebrugh's Sets for Mathematics, they write
It is a theorem [MM92] that a topos is well-pointed if and only if it is Boolean, two-valued, and supports split.
[MM92] is a reference to ...
11
votes
0answers
188 views
Lawvere's Perugia Notes
A PDF version of Lawvere's Perugia Notes (Theory of Categories over a Base Topos) can be found here: https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf
That PDF ...
7
votes
0answers
144 views
Internal logic of the small étale topos of an algebraic variety
If we consider the internal logic of the small étale topos of an algebraic variety, is the variety's geometry reflected in it? For instance, how different are the internal logics for the étale topoi ...
15
votes
1answer
869 views
Reconstruct a variety from its crystalline topos
Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.
Can we reconstruct $X$ from its small crystalline topos $((X/...
5
votes
0answers
145 views
Surprising examples of functors which preserve cofiltered limits but not all limits?
Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits?
I'm not quite sure what "surprising" means, but I think ...
13
votes
2answers
675 views
Can one associate a “nice” topos to a von Neumann algebra?
The question here inspires my present question.
Reyes proves here that the contravariant functor Spec from the category of commutative rings to the category of sets cannot be extended to the category ...
2
votes
1answer
113 views
Constructivity of two problems on a standard simplex?
Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and non-negative and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always ...
