All Questions
Tagged with topos-theory locales
21 questions
9
votes
1
answer
369
views
G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
user513306
12
votes
4
answers
507
views
Localic or topos-theoretic definition of $\operatorname{Spec}$
Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes&...
- 253
9
votes
1
answer
545
views
Is there a good theory of 2-locales?
Topological spaces and locales are two closely related notions meant to capture the concept of an abstract space, the latter of which admits a certain "noncommutative" generalisation known ...
- 11.8k
2
votes
0
answers
101
views
Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
8
votes
2
answers
754
views
Every Grothendieck topos can be built from localic topoi
Theorem 2 in these notes[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the theorem of Joyal and ...
- 83
6
votes
1
answer
215
views
Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary)
This is a crosspost from math.stackexchange.
A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\...
- 63
2
votes
0
answers
113
views
Explicit description of the canonical $\pi_G: \mathrm{Sh}(G_0)\to B_S(\mathbf{G})$
Given a localic groupoid $\mathbf{G} = (G_1\overset{d_0}{\underset{d_1}{\rightrightarrows}}G_0)$ and letting $B\mathbf{G}$ denote its classifying topos, I'm looking for a explicit description of the ...
5
votes
0
answers
175
views
Within pointless topology inside of choiceless constructivism, prove that division is possible
In Frank Waaldijk's paper on the foundations of constructive analysis, Waaldijk shows that various definitions of "continuous function" for functions of the form $f: \mathbb R \to \mathbb R$ ...
- 4,943
9
votes
1
answer
505
views
Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?
Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{...
- 395
9
votes
1
answer
356
views
What's the localic reflection of a presheaf topos?
$\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.
What's the localic reflection of $\Psh ...
- 1,083
12
votes
0
answers
432
views
What does the localic reflection of a classifying topos classify?
Let $\mathbb{T}$ be a geometric theory. Let $\mathrm{Set}[\mathbb{T}]$ be its classifying topos, such that geometric morphisms from any (cocomplete) topos $\mathcal{E}$ into $\mathrm{Set}[\mathbb{T}]$ ...
- 3,312
5
votes
1
answer
200
views
Topological regularity for toposes
A topological space $X$ is regular if a point not belonging to a closed set can be separated from it by disjoint opens. Equivalently, this means if $x\in U$ for an open set $U$, then there are opens $...
- 66.8k
23
votes
2
answers
2k
views
Locales as geometric objects
There is the following analogy:
$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$
Here, ...
- 5,482
3
votes
0
answers
454
views
Topos Theory, internal Heyting Algebra
Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.
If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...
- 159
15
votes
1
answer
1k
views
The real numbers object in Sh(Top)
If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...
- 66.8k
0
votes
0
answers
230
views
Is there a translation invariant measure on an infinite dimensional space 'without points'?
This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...
- 2,369
12
votes
1
answer
532
views
Are $\infty$-topoi determined by their localic points ?
Hello !
If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
- 42.4k
9
votes
2
answers
590
views
Given a Grothendieck topos, what does its localic groupoid look like? [duplicate]
Possible Duplicate:
Toposes (topoi) as classifying toposes of groupoids
For example, if a topos E is the object classifier, or the preseaf topos on a small category C, is there a way of ...
3
votes
1
answer
860
views
Counterexemple to Urysohn's lemma in a topos without denombrable choice ?
Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...
- 42.4k
25
votes
0
answers
1k
views
$\infty$-topos and localic $\infty$-groupoids?
It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting ...
- 42.4k
8
votes
6
answers
2k
views
Stone Spaces, Locales, and Topoi for the (relative) beginner
I am currently reading Vickers' text "topology via logic" and Peter Johnstone's "stone spaces", and I understand the material in both of these texts to pertain directly to constructions in elementary ...
- 1,539