17
votes
Accepted
What's the point of a point-free locale?
A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras,
which shows that the following categories are equivalent:
The ...
- 37.8k
11
votes
Accepted
Product of topological spaces and product of corresponding locales
Your map $f$ is known to be an injective dense localic map. See, for example, Proposition 4.2.2 in [1]. In general, it isn't an isomorphism. The reason for this is that $\Omega(X \times_t Y)$ is ...
- 621
4
votes
Dissolution of a topos
I'm not aware of litterature on this, but this is something I have thought about several years ago and never ended-up using or publishing. What is below is me trying to remind myself how it works - ...
- 42.4k
3
votes
Preimage of a sublocale by a morphism of locales: description by nucleus?
Here is I think a counter-example to the precise proposed formula in the question.
Take $X= \mathbb{Q}$ with the discrete topology, with $Y = \mathbb{R}$ withe its usual topoly and the map $f:X \to Y$ ...
- 42.4k
1
vote
Accepted
"Locally compact"-ly generated topological spaces
If $X$ is locally compactly generated then $X$ is compactly generated because every locally compact space is compactly generated.
So given $f:X \to Y$ a map such that $f\circ i$ is continuous for ...
- 42.4k
1
vote
Accepted
What is the power of the “anti-halting” oracle?
The set $K_1 := \bigcap_{B\subseteq\mathbb{N}} (H_1(B) \Rrightarrow B)$ is inhabited, that is, the “anti-halting oracle” is trivial, i.e., can be simulated with a Turing machine (consequently, as ...
- 32.5k
1
vote
Accepted
Computing the Heyting operation on the frame of nuclei
First, every nucleus is the join of those of the form $j^x\land j_y$.
Namely,
$$
j=\bigvee_xj^x\land j_{jx}
$$
Hm, this is too confusing. Let me change notation and write: $u_a$ instead of $j^a$; $v_a$...
- 18.4k
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