Questions tagged [domain-theory]
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12 questions
3
votes
1
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375
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Injective envelopes, constructively?
Question 0:
How do injective envelopes work in constructive mathematics?
For example,
Question 1: How strong is it to assert internally that there are enough injectives (in the category of sets, say)? ...
- 68.1k
8
votes
1
answer
204
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Two questions on forests with distinguished children
Let
$$X_0 \mathop \rightleftharpoons^{s_0}_{r_0} X_1 \mathop \rightleftharpoons^{s_1}_{r_1} X_2 \mathop \rightleftharpoons^{s_2}_{r_2} \cdots$$
be a diagram (in some bicomplete category) such that $...
- 2,205
7
votes
2
answers
520
views
Cauchy completeness in ordered families of equivalences, constructively
An ordered family of equivalences (OFE) is a set $X$ equipped with an $\omega$ indexed family of equivalence relations $\approx_i$ that are increasingly fine-grained in that if $x \approx_{i+1} y$ ...
- 1,081
2
votes
0
answers
185
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Rings with algebraic Zariski spectrums
A topological space is coherent if it has a topology generated by (quasi)compact opens. A compact open is coprime if it can't be represented as a non-trivial union of compact opens. A space is ...
- 813
6
votes
1
answer
351
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What's the relationship between the Zariski and Scott topologies on the (reverse-ordered) spectrum?
I don't know anything about algebraic geometry. I was bored at work, reading nLab, and noticed that the Zariski topology and Scott topology are vaguely similar. Strictly $T_0$, and almost never ...
- 411
14
votes
2
answers
605
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Synthetic differential / conformal geometry of Lorentzian manifolds?
Let $M$ be a sufficiently nice Lorentzian manifold of dimension $\geq 3$. It's known [1] (see also [2]) that the differential and even conformal structure of $M$ is completely encoded in the causal ...
- 68.1k
4
votes
1
answer
179
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Example of a bicontinuous poset which is not jointly bicontinuous?
Recall that a poset $P$ is said to be continuous if, for every $p \in P$, the set $\{q \in P \mid q \ll p \}$ is directed with supremum $p$. Here $q \ll p$ is the "way below" relation (see ...
- 68.1k
1
vote
1
answer
285
views
Hyperplane separation of a concave functional and a point, in domain theory
Problem:
Let $D$ be an $\omega$-BC domain, and $[D\to[0,\infty]]$ be the space of Scott-continuous nonnegative functions on $D$, equipped with the obvious ordering and the Scott-topology.
EDIT: I don'...
- 353
3
votes
0
answers
184
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Closure properties of models of synthetic domain theory
I would like to know what topos models of synthetic domain theory are closed under. Here are some questions that I would be interested in an answer to:
Suppose $\mathbf{E}$ is an SDT-topos, and let $\...
3
votes
1
answer
775
views
What is the cardinality of Dana Scott's $D_{\infty}$?
In the 1960's, Dana Scott constructed the domain $D_{\infty}$ which has the property
$D_{\infty} \cong D_{\infty}{}^{D_{\infty}}$.
Its construction is based on a cumulative hierarchy of infinite ...
- 588
7
votes
0
answers
244
views
On thinking of spacetime as a local Scott domain
An observation of Martin and Panangaden links the study of Lorentzian manifolds and the semantics of programming languages via the theory of Scott domains.
Background:
Recall that if $M$ is a time-...
- 68.1k
3
votes
0
answers
168
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When can all elements of $[A\to B]$ can be represented as computable functions?
(crosspost from math stackexchange)
While working through Barendreght's book on the Lambda Calculus, and Abramsky's notes on Domain Theory, I had the following realization:
It's often stated that ...
- 353