All Questions
Tagged with model-theory ct.category-theory
78
questions
11
votes
1
answer
665
views
On the classification of second-countable Stone spaces
Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent:
$X$ is second countable
$X$ is metrizable
$X$ has countably many clopen subsets
$X$ is an ...
- 62.6k
11
votes
1
answer
436
views
Are flat functors out of a finite category necessarily finite?
Note: I've originally asked this question on math stack exchange, but I have learnt that this is the better place to ask for research level questions, so I have deleted the original question there.
...
- 113
2
votes
1
answer
557
views
Logical content of Gauss's Lemma (arithmetic)
In the context where $a$, $b$, $c$ are integers we have $(a \mid bc, a\land b = 1) \Rightarrow a\mid c$. This result is called Gauss's Lemma in French Highschool.
It is well known that (Steve Awodey, ...
- 29
5
votes
0
answers
227
views
Classical first-order model theory via hyperdoctrines
I have been reading this discussion by John Baez and Michael Weiss and I find this approach to model theory using boolean hyper-doctrines very interesting. One of their goal was to arrive at a proof ...
- 3,084
7
votes
0
answers
290
views
A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$
Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have
$M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
- 20.7k
4
votes
2
answers
261
views
Symmetric monoidal functors from powers of the natural numbers to Set
Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable infinite sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between ...
- 553
1
vote
1
answer
131
views
For a pure-injective module $M$ does the property "$\operatorname{Hom}(-,M)$ is surjective" commute with certain limits?
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\...
- 1,023
1
vote
1
answer
97
views
Choosing a net of projections from a given collection
Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...
- 1,807
5
votes
1
answer
2k
views
Do bijections from the natural numbers satisfy the Peano axioms? [closed]
While thinking of natural numbers as anything that satisfies the Peano axioms, I was left wondering, what if I take the successor function $S(x)$ to be anything other than $x\to x+1$?
Some examples ...
- 129
3
votes
1
answer
266
views
Ultra*powers* in the category of structures and elementary embeddings
This is based on a few previous questions.
Can one characterize ultrapowers in the category of L-structures (modeling a fixed complete theory, say) and elementary embeddings?
Previous posts showed ...
- 151
6
votes
1
answer
553
views
Ultraproducts in the category of structures and elementary embeddings
A previous question on the categorical nature of ultraproducts had great answers, mostly categorically characterizing ultraproducts in the category of $L$-structures and homomorphisms for a fixed ...
- 151
12
votes
1
answer
413
views
The scope of a "strong Cantor-Bernstein" property
This question is of course related to this earlier MO question, but I don't believe is answered by the posts there.
My favorite proof of the Cantor-Schroeder-Bernstein theorem actually establishes ...
- 20.7k
17
votes
3
answers
2k
views
Recommendations to learn about the use of toposes in logic?
I'd like to learn about the use of toposes in logic. The "logic" side I know quite well, but of the "topos" side I am totally ignorant.
Which books/articles (formal and/or casual) ...
- 1,005
13
votes
0
answers
317
views
When does HSP reduce to SPH?
This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
- 17.8k
5
votes
1
answer
197
views
When is an elementary subclass reflective?
Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more ...
- 424