All Questions
Tagged with topos-theory gn.general-topology
20 questions
9
votes
1
answer
456
views
Topos notions coming from topology and uniqueness of generalizations
Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call ...
- 1,347
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
2
votes
0
answers
156
views
Do Grothendieck topoi with enough points satisfy the fan theorem internally?
Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it.
This leads one to conjecture a ...
- 1,947
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
7
votes
1
answer
465
views
When is a basis of a topological space a Grothendieck pretopology?
Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...
- 1,947
8
votes
1
answer
1k
views
What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 5,230
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
$$\mathrm{Sh}\colon\mathbf{...
- 395
18
votes
2
answers
617
views
In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?
For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
- 32.5k
4
votes
0
answers
134
views
What does the Grothendieck topos tell us about the homotopy type of a space?
Let $M_1$, $M_2$ be two closed connected topological manifolds. We can consider the small sites of open coverings of them, and the categories of sheaves on these sites.
what can we say about $M_1$ ...
user140978
13
votes
1
answer
570
views
Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
- 18.4k
17
votes
2
answers
2k
views
Foundations of topology
I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here.
Also some time ago I read ...
- 1,190
17
votes
7
answers
1k
views
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 ...
3
votes
0
answers
431
views
Bohr topos and quantization
Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...
- 585
11
votes
4
answers
2k
views
Embedding Theorem for topological spaces, and in general
There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...
- 1,838
6
votes
3
answers
1k
views
Is there a category of topological-like spaces that forms a topos?
The category of convergence spaces generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a ...
- 2,083
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
53
votes
3
answers
8k
views
Grothendieck's manuscript on topology
Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
2
votes
1
answer
243
views
Induced pretopologies on sSet
Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
- 35.5k
6
votes
1
answer
765
views
Are finite colimits of topological spaces stable under pull-back?
The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a Grothendieck topos and ...
- 27.5k
5
votes
0
answers
336
views
Defining a topology by means of closed subsets in a topos
In the following we fix a topos. I'll speak of sets instead of objects and of subsets instead of subobjects.
Let $X$ be a set and assume $F$ is a set of subsets of $X$ that contains $\emptyset, X$, ...
- 63.1k