All Questions
8
questions
7
votes
1
answer
398
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 ...
- 1,803
8
votes
1
answer
806
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 ...
- 4,760
7
votes
1
answer
439
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{...
- 375
17
votes
2
answers
549
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 ...
- 28.7k
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 ...
- 1,513
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 ...
- 2,023
3
votes
1
answer
811
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 ...
- 39.4k
2
votes
1
answer
242
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 ...
- 33.2k