All Questions
Tagged with gn.general-topology ct.category-theory
183
questions
37
votes
1
answer
1k
views
What is the meaning of this analogy between lattices and topological spaces?
Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...
- 41.5k
1
vote
0
answers
178
views
$\mathbb E$-descent maps in topological spaces in terms of different sites?
The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...
- 11
9
votes
0
answers
279
views
Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?
For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff $f^{-1}...
- 59k
55
votes
3
answers
3k
views
Duality between compactness and Hausdorffness
Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
- 1,691
6
votes
1
answer
443
views
Universal covering and double cover functors
Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...
- 319
6
votes
1
answer
120
views
Characterisations of closed embeddings in $Top_1$?
Let $Top_1$ be the category of topological spaces which are $T_1.$
I am curious as to whether there is a categorical definition of what a closed embedding is in this environment. With a ...
- 61
9
votes
1
answer
466
views
Is every locally compactly generated space compactly generated?
[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...
5
votes
0
answers
70
views
Does the $D$-property have universal objects?
A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...
- 44.5k
1
vote
0
answers
124
views
Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
- 369
14
votes
2
answers
1k
views
Which sequential colimits commute with pullbacks in the category of topological spaces?
This question was asked on math.stackexchange.com without a reaction.
Given diagrams of topological spaces
$$X_0\rightarrow X_1\rightarrow\ldots$$
$$Y_0\rightarrow Y_1\rightarrow\ldots$$
$$Z_0\...
- 141
20
votes
2
answers
1k
views
The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,324
3
votes
1
answer
295
views
Exponential locales and a pointless version of the compact-open topology?
TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an ...
- 363
2
votes
0
answers
175
views
Monadicity of profinite algebras
We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples.
In case that I was ...
- 121
4
votes
2
answers
571
views
The classifying space of an infinite totally ordered set is contractible
I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...
35
votes
2
answers
5k
views
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
- 2,954
27
votes
3
answers
1k
views
Possible categorical reformulation for the usual definition of compactness
Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...
- 373
1
vote
0
answers
229
views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,424
1
vote
0
answers
162
views
The category of discontinuous Banach spaces
A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
- 280
3
votes
2
answers
684
views
Topological retraction vs categorical retraction
Let $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a
topological retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have $...
- 44.5k
3
votes
2
answers
547
views
Is the defining bijection for a pullback of topological spaces a homeomorphism?
I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map
$$Top(T,P) \rightarrow Top ...
4
votes
1
answer
197
views
When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?
Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...
- 1,961
8
votes
2
answers
655
views
Topological characterization of injective metric spaces
Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \...
- 7,424
12
votes
1
answer
2k
views
Reference request: Book of topology from "Topos" point of view
Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
- 535
37
votes
13
answers
4k
views
Continuous relations?
What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I ...
- 852
24
votes
0
answers
899
views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on $...
- 8,327
8
votes
3
answers
936
views
Do any Stone-like dualities have some self-dualities hidden inside them?
This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
- 17.3k
2
votes
0
answers
177
views
A categorical analogue of Debreu's independent factors theorem
Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
12
votes
4
answers
1k
views
Categorical Construction of Quotient Topology?
The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies ...
- 1,279
9
votes
0
answers
360
views
Is there Ultracoproduct-like construction for topological spaces in general?
In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
- 191
79
votes
5
answers
5k
views
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
- 114k
8
votes
1
answer
599
views
What should the morphisms in the Category of Directed Sets be?
Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...
- 352
14
votes
3
answers
688
views
Is there a monad on Set whose algebras are Tychonoff spaces?
Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.
Is the category of Tychonoff spaces also monadic over Set?
- 3,123
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
15
votes
6
answers
3k
views
Giving $\mathit{Top}(X,Y)$ an appropriate topology
$\DeclareMathOperator\Top{\mathit{Top}}$I am not sure if its OK to ask this question here.
Let $\Top$ be the category of topological spaces. Let $X,Y$ be objects in $\Top$.
Let $F:\mathbb{I}\...
- 1,025
3
votes
2
answers
679
views
Finitely cocomplete categories of compact Hausdorff spaces
Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 ...
- 6,129
0
votes
3
answers
379
views
Existence of a Sub-Category of the Category of Topological Spaces
My question start with the following observations:
If you have a finite number of topological spaces $X_1, \dots , X_n$ you can define a space that is the disjoint union of its $\sqcup_{i=1}^n X_n=Y$....
- 325
16
votes
1
answer
2k
views
Pullbacks as manifolds versus ones as topological spaces
My question is: Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks?
Detailed explanation is following.
A pullback is defined as a manifold/topological space satisfying a universal ...
- 801
17
votes
3
answers
1k
views
Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 8,322
11
votes
2
answers
2k
views
Category of topological spaces with open or closed maps
Consider the category whose objects are topological spaces and whose morphisms are the open maps (or closed maps, open continuous maps, closed continuous maps ... that is, one whose isomorphisms are ...
- 2,422
4
votes
1
answer
249
views
function space in comma category
Let TOP be a category of topological spaces and B be an object of TOP. Is there a notion of function space in the comma category TOP/B.
- 41
6
votes
2
answers
476
views
A continuous notion of realizability
I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...
- 19k
6
votes
0
answers
320
views
Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
- 19.8k
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
4
votes
1
answer
1k
views
Abstract definition of properly discontinuous action
A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract ...
- 1,181
8
votes
2
answers
1k
views
Category of Uniform spaces
I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and cartesian closed. Can anyone comfirm or deny, ...
- 542
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
6
votes
1
answer
742
views
When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
- 712
1
vote
0
answers
430
views
Universal Hausdorff Space [duplicate]
Possible Duplicate:
Largest Hausdorff quotient
Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$...
user2529
8
votes
2
answers
418
views
Is the category of quotient of countably based topological spaces cartesian closed ?
In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\...
- 774
3
votes
1
answer
480
views
Why the category of core-compact spaces with continuous maps is not cartesian closed ?
According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:
A topological space $X$ is ...
- 774