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Results tagged with ct.category-theory
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user 61536
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
12
votes
Accepted
Analogue of Urysohn metrization for Lawvere metric spaces?
According to this SE-post, a Lawvere metric on a set $X$ is a function $d:X\times X\to[0,+\infty]$ satisfying two axioms:
$d(x,x)=0$ and
$d(x,z)\le d(x,y)+d(y,z)$
for all $x,y,z\in X$.
Then the fo …
- 41.8k
11
votes
1
answer
959
views
Why do elementary topoi have pullbacks?
In the book of Szabo "Algebra of Proofs", Definition 13.1.9 introduces an elementary topos as a cartesian closed category with a subobject classifier. On the other hand, many other sources including J …
- 41.8k
7
votes
2
answers
604
views
What is the name for a set endowed with a Lipschitz structure?
I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the Lipsc …
- 41.8k
7
votes
2
answers
2k
views
What is a good definition of a mathematical structure?
At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist app …
- 41.8k
6
votes
Is there a category of topological spaces such that open surjections admit local sections?
Maybe this is not quite satisfactory answer, but the category of zero-dimensional Polish spaces and their continuous maps has the required property: each open continuous map between Polish zero-dimens …
- 41.8k
4
votes
1
answer
378
views
What does play the role of a subobject classifier for quotient objects?
It is known that in the category of sets the dualization of the notion of a subobject classifiers does not work because the only object admitting a morphism into an initial object is the empty set.
B …
- 41.8k
4
votes
1
answer
173
views
Categories admitting singleton-classifiers and characterization of the category $\mathbf{Set}$
Trying to characterize categories equivalent to the category of sets, I have discovered (for myself) that instead of requiring that the coprojection morphism $\mathsf{true}:1\to \Omega=1\sqcup 1$ is a …
- 41.8k
3
votes
What kinds of operations are well-defined when working with sets, classes, conglomerates, an...
Maybe too late, but only today I have seen this question posed 10 years ago.
Indexed families of (proper) classes can be legally defined in NBG: an indexed family of classes $(X_\alpha)_{\alpha\in A}$ …
- 41.8k
2
votes
1
answer
131
views
The separability of superextensions
The superextension $\lambda X$ of a compact Hausdorff space $X$ is the space of maximal linked systems of closed subsets of $X$, endowed with the Vietoris topology inherited from the double hyperspace …
- 41.8k
1
vote
Mathematics Roadmap
Good question. 100 years ago it was much easier to answer it. For example, the "map" of mathematics drawn by Janiszewski in 1915 in the book "Poradnik dla samoukow" looked as follows:
Now everything b …
- 41.8k
-3
votes
1
answer
233
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F( …
- 41.8k