All Questions
Tagged with gn.general-topology ct.category-theory
196 questions
13
votes
3
answers
670
views
How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
- 12.4k
9
votes
1
answer
456
views
Topos notions coming from topology and uniqueness of generalizations
Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call ...
- 1,347
7
votes
1
answer
134
views
Universally closed implies proper for locales
It is well known that:
Theorem.
For a locale (resp. topological space) $X$, the following are equivalent:
$X$ is compact, i.e. every open cover of $X$ has a finite subcover.
For every locale (resp. ...
- 15.9k
18
votes
0
answers
323
views
The analogy between dualizable categories and compact Hausdorff spaces
Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{...
- 2,303
2
votes
0
answers
104
views
When do filtered colimits commute with finite products in Top
It is well known that filtered colimits commute with finite products (more generally any finite limit). This is not the case in general in Top due to Top not being cartesian closed. My question is is ...
- 41
13
votes
0
answers
260
views
Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
5
votes
1
answer
311
views
Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
- 16.4k
2
votes
1
answer
179
views
Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
- 615
11
votes
1
answer
755
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 ...
- 63.9k
2
votes
1
answer
132
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
5
votes
1
answer
270
views
Are Euclidean spaces $\Delta$-generated?
From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$.
However, the ...
- 53
4
votes
1
answer
223
views
Is every compact, sober, second-countable space the image of $2^\omega$?
As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$?
As a further bonus, can we strengthen "image" to "quotient"?
My motivation for ...
- 3,612
5
votes
1
answer
191
views
Monoidal topology and coarse spaces
Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?
- 3,816
7
votes
0
answers
271
views
Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
- 519
5
votes
0
answers
131
views
Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
- 63.9k
1
vote
1
answer
344
views
Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
3
votes
1
answer
550
views
Do CGWH spaces form an exponential ideal in Condensed Sets?
If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
user103549
6
votes
1
answer
457
views
Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every ...
- 63.9k
8
votes
1
answer
181
views
Stone-topological/profinite equivalence for quandles
A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...
- 253
4
votes
1
answer
233
views
Profinite groups with isomorphic proper, dense subgroups are isomorphic
I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
- 253
6
votes
0
answers
246
views
Making the analogy of finiteness and compactness precise
If one asks about the intution behind compact topological spaces, most often one will hear the mantra
“Compactness of a topological space is a generalisation of the finiteness of a set.”
For example,...
- 1,474
18
votes
0
answers
1k
views
"Next steps" after TQFT?
(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)
Recently, I've been ...
- 377
4
votes
0
answers
231
views
path category and classifying space
Let $\mathbf{Top}$ be the category of topological spaces and continuous maps, and $\mathbf{Cat}$ be the category of small categories and functors.
There is a path functor $\mathcal{P}:\mathbf{Top}\to \...
- 153
8
votes
0
answers
172
views
The pro-discrete space of quasicomponents of a topological space
Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...
- 15.9k
10
votes
1
answer
448
views
Do compactly generated spaces have a more direct definition?
Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first?
Weakly Hausdorff sequential spaces ...
- 1,947
3
votes
0
answers
101
views
Constructively valid reference for the soberness of discrete spaces and points of a locale coproduct
I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of ...
- 32.5k
9
votes
1
answer
734
views
Does the category of locally compact Hausdorff spaces with proper maps have products?
nlab presents a proof that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since ...
- 295
7
votes
1
answer
384
views
Compact Hausdorff spaces as a cocompletion of profinite sets
It is well-known that the category CH of compact Hausdorff spaces has a strong categorical flavor (e.g. Properties of the category of compact Hausdorff spaces, which includes Manes' theorem asserting ...
- 2,040
14
votes
3
answers
2k
views
Is there a universal property characterizing the category of compact Hausdorff spaces?
This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces
To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\...
- 2,303
6
votes
1
answer
198
views
Subobject classifier in $\mathsf{Top}^{D^{\text{op}}}$?
Let $D$ be a small category. Does the category of diagrams $\mathsf{Top}^{D^{\text{op}}}$ have a classifier of (strong?) subobjects? I tried following the "sieve construction" for the ...
- 677
7
votes
1
answer
268
views
Are closed embeddings characterized by a left lifting property in the category of topological spaces?
It is well-known and easy to check that a continuous map between topological spaces is an embedding if and only if it has the LLP with respect to $A \to *$ and $B \to *$ where $A$ is the two-point ...
- 7,656
3
votes
1
answer
190
views
An extension of Stone duality
First let me recall Stone duality in terms of propositional logic.
Let $L$ and $K$ be propositional signatures (i.e., sets of propositional variables). Let $T$ be a propositional theory over $L$ and $...
- 133
5
votes
0
answers
204
views
What are all of the topological (commutative) monoid structures on a closed interval?
Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.
Question 1: What are all of the topological commutative ...
- 63.9k
6
votes
1
answer
484
views
Why finitely presentable objects in Top need to be discrete?
In Locally Presentable and Accessible Categories, page 12 (10),
A topological space is finitely presentable in $\mathbf{Top}$, the category of topological spaces and continuous functions, iff it is ...
- 193
4
votes
1
answer
817
views
Adjunction between topological spaces and condensed sets
I am trying to prove that the functor
\begin{align*}
\mathrm{Top} &\longrightarrow \mathrm{Cond}(\mathrm{Set}) \\
X &\longmapsto \underline{X}
\end{align*}
admits a left adjoint and it is the ...
- 1,485
7
votes
1
answer
353
views
Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 173
7
votes
1
answer
465
views
When is a basis of a topological space a Grothendieck pretopology?
Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...
- 1,947
9
votes
0
answers
211
views
Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
8
votes
1
answer
1k
views
What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 5,230
8
votes
1
answer
360
views
Is Hausdorffness a categorical property in the category of locally convex spaces?
I want to characterize Hausdorffness of a locally convex space only using categorical terms of the additive category LCS of locally convex spaces and continuous linear maps, i.e., terms like mono- or ...
- 16.4k
15
votes
3
answers
1k
views
Why it is convenient to be cartesian closed for a category of spaces?
In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
- 9,111
9
votes
1
answer
505
views
Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?
Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{...
- 395
9
votes
2
answers
438
views
What are projective locales / injective frames?
Judging by the compact regular case, and more generally the spatial case, regular projectivity of locales, resp. regular injectivity of frames, must have something to do with $\neg p\lor\neg\neg p$ ...
- 18.4k
2
votes
1
answer
173
views
(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces
I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a ...
- 898
3
votes
1
answer
395
views
Closed embedding into a normal Hausdorff space and left lifting property
I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a ...
- 3,901
4
votes
0
answers
242
views
Inductive limit of inclusions
Let $(\Lambda, \le)$ be a directed system and $\{ X_{\alpha} \}_{\alpha \in \Lambda}$ be a family of topological spaces indexed by $\Lambda$ such that $X_{\alpha} \subseteq X_{\beta}$ whenever $\alpha ...
- 183
14
votes
2
answers
761
views
Is there a large colimit-sketch for topological spaces?
Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
- 63.1k
5
votes
1
answer
225
views
Closure of the diagonal is an equivalence relation
Let $X$ be a topological space and $\overline{\Delta_X} \subseteq X \times X$ the closure of its diagonal.
Then $\overline{\Delta_X}$ is the graph of an equivalence relation on $X$.
This statement can ...
- 51
8
votes
1
answer
278
views
What is known about these "explicitly represented" spaces?
Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here.
The standard approach ...
- 3,612
0
votes
1
answer
188
views
Sober spaces vs. spatial frames-a big picture
For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...
- 9,330