All Questions
Tagged with gn.general-topology ct.category-theory
183
questions
15
votes
1
answer
455
views
What are the algebras for the ultrafilter monad on topological spaces?
Motivation: Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), ...
- 59k
7
votes
0
answers
253
views
When is the exponential of a map proper?
Let $X$ be a compact Hausdorff space. Then if $f: A \to B$ is a map between discrete spaces, the induced map $f^\ast: X^B \to X^A$ is proper.
Question: Are there other classes of map $f: A \to B$ ...
- 59k
5
votes
1
answer
606
views
Can $L^1_{loc}$ be represented as colimit?
Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
- 5,001
1
vote
0
answers
115
views
Stone duality- a modification
Let $2$ be the discrete topological space with two elements. For a map of sets
$$\beta : X \times Y \rightarrow 2 $$
We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest ...
- 4,647
36
votes
1
answer
3k
views
Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
- 59k
17
votes
1
answer
461
views
Combination topological space and locale?
The traditional theory of topological spaces (as formalized by Bourbaki) starts with a set of points, then builds a structure on that. In contrast, the theory of locales starts with a frame of opens (...
- 2,584
4
votes
1
answer
354
views
Functor from rings into compact Hausdorff spaces
There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}_2]_{...
- 4,647
9
votes
1
answer
438
views
On the universal property for interval objects
In his lecture, The Categorical Origins of Lebesgue Measure, Professor Tom Leinster mentions the following theorem:
Theorem 1: (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with ...
- 4,647
4
votes
0
answers
477
views
A slightly canonical way to associate a scheme to a Noetherian spectral space
Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
user139951
9
votes
0
answers
214
views
Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
- 4,647
3
votes
1
answer
239
views
Category of continuous self maps
Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous ...
- 2,572
7
votes
2
answers
255
views
The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
- 6,641
40
votes
2
answers
2k
views
Ultrafilters as a double dual
Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known:
$X$ canonically embeds into $\beta X$ (by taking principal ultrafilters);
If $X$ is finite, then there ...
- 12.1k
36
votes
3
answers
3k
views
What is the structure preserved by strong equivalence of metrics?
Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
- 4,489
14
votes
1
answer
531
views
"Scott completion" of dcpo
If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...
- 39.4k
5
votes
1
answer
292
views
Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?
We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{...
- 1,576
5
votes
0
answers
156
views
For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?
Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.
For which ...
- 10.3k
3
votes
1
answer
93
views
sequences of iterated orthogonals (lifting property) in a category
I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...
- 99
5
votes
1
answer
230
views
Finally dense implies dense
I am reading the article "A convenient category for directed homotopy" by Fajstrup and Rosicky and I have a doubt about the proof of Proposition 3.5. The setting is the following:
let $\cal{C}$ be a ...
- 303
4
votes
1
answer
132
views
Is the category of inclusion prespectra bicomplete?
Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete?
I should probably specify that by inclusion prespectra, I mean prespectra such that the ...
- 765
2
votes
1
answer
209
views
Commutation of filtered colimits and finite limits in $\mathbb{CGWH}$
Do filtered colimits and finite limits (in particular pullbacks) commute in the category of compactly generated weak Hausdorff spaces?
- 21
6
votes
0
answers
146
views
Spatiality of products of locally compact locales
In Johnstone´s Sketches of an Elephant Volume 2, page 716,
lemma 4.1.8 states that for spatial locales $X$ and $Y$ with $X$ locally compact then the locale product $X\times Y$ is spatial.
Is this ...
- 159
10
votes
1
answer
281
views
Analogue of Urysohn metrization for Lawvere metric spaces?
Urysohn proved that any regular, Hausdorff, second-countable space $X$ is metrizable, i.e. there exists a metric space whose underlying topological space is $X$. But what if we ask the same question ...
- 8,327
5
votes
1
answer
611
views
Exponential law w.r.t. compact-open topology
It is well-known that if a topological space $Y$ is
locally compact (not necessarily Hausdorff),
then the map
$$
\operatorname{Hom}(X \times Y, Z) \to
\operatorname{Hom}(X, Z^Y)
$$
(here we use the ...
- 368
15
votes
1
answer
815
views
Homotopy pullback of a homotopy pushout is a homotopy pushout
Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
- The right square is a homotopy pushout and
- all the squares in the middle are ...
- 181
6
votes
3
answers
381
views
Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze ...
7
votes
2
answers
589
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 ...
- 40.2k
11
votes
1
answer
741
views
Colimits, limits, and mapping spaces
It is true that in the category of topological spaces
$ \mathrm{Map}(\underset{i\in I}{\mathrm{colim}}\, X_i, Y)\cong
\underset{i\in I}{\mathrm{lim}}\,\mathrm{Map}(X_i,Y)$ ? Here mapping spaces are ...
- 1,695
13
votes
2
answers
508
views
Constructive proofs of existence in analysis using locales
There are several basic theorems in analysis asserting the existence of a point in some space such as the following results:
The intermediate value theorem: for every continuous function $f : [0,1] \...
- 4,340
11
votes
1
answer
2k
views
What are compact objects in the category of topological spaces?
Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits.
On the other hand, ...
- 26.5k
3
votes
1
answer
263
views
Adjoints for the functor ${\bf Top}\to {\bf Conv}$
Let $X$ be a set and let $\Phi(X)$ denote the collection of filters on $X$. For $x\in X$ we denote by $P_x$ the filter $P_x=\{A\subseteq X:x\in A\}$. A convergence space is a pair $(X,\to)$, where $X$ ...
- 44.5k
3
votes
1
answer
222
views
Is the category of convergence spaces cartesian-closed?
Convergence spaces are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ cartesian-closed?
- 44.5k
2
votes
1
answer
158
views
Adjoints of the interval topology functor
Given a poset $(P,\leq)$ the interval topology $\tau_i(P)$ on $P$ is generated by
$$\{P\setminus{\downarrow x} : x\in P\} \cup \{P\setminus{\uparrow x} : x\in P\},$$
where $\downarrow x = \{y\in P: y\...
- 44.5k
5
votes
0
answers
311
views
What is the local structure of a fibration?
It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled.
Here I'd like ...
- 59k
3
votes
0
answers
110
views
Functorial description of irreducibility of topological space?
This is a crosspost of this MSE question.
A topological space is connected if it's not the coproduct of two non-trivial spaces. Equivalently, it is connected if the copresheaf it represents preserves ...
- 10.3k
11
votes
4
answers
1k
views
What was Burroni's sketch for topological spaces?
In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
- 2,859
2
votes
2
answers
405
views
Topological space (or math structure more generally) without encoding as set
Given the historical development of modern mathematics, everything is ultimately encoded as a set (possibly with some additional structure, also encoded as set(s) ). For example, a topological space ...
- 233
7
votes
1
answer
398
views
Does the forgetful functor $\mathbf{Comp} \rightarrow \mathbf{Top}$ have a left-adjoint?
The forgetful functor $\mathbf{CompHaus} \rightarrow \mathbf{Top}$ from compact Hausdorff spaces to topological spaces famously has a left-adjoint, the Stone-Cech compactification.
Question. Does ...
- 3,683
10
votes
2
answers
469
views
Is there a notion of "space" such that vector bundles can be understood in this way?
Is there a notion of "space" satisfying the following requirements?
Spaces form (at least) a category; morphisms between spaces are called "continuous maps."
Every topological space is a space, and ...
- 3,683
4
votes
1
answer
326
views
Categorical Description of Open Subspaces
In Top, the monos are the injective maps and the regular monos are the subspace inclusions. Is there a (similarly pithy) categorical description for the open subspace inclusions?
- 141
2
votes
3
answers
480
views
examples of lifting properties
A number of seemingly unrelated elementary notions can be defined uniformly with help of (iterated) Quillen lifting property
(a category-theoretic construction I define below) "starting" to a single (...
- 41
10
votes
1
answer
631
views
Topology from the viewpoint of the filter endofunctor
Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of ...
- 3,683
71
votes
1
answer
2k
views
Dualizing the notion of topological space
$\require{AMScd}$
Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements ...
- 4,647
8
votes
1
answer
253
views
Algebraic characterization of convergence spaces
Is there an algebraic characterization of convergence spaces similar to Barr's characterization of topological spaces as lax algebras for the ultrafilter monad? I'm also curious about the same ...
- 3,222
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.3k
3
votes
0
answers
130
views
Duality for continuous lattices based on [0, 1]
A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the ...
- 133
23
votes
5
answers
2k
views
The "right" topological spaces
The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical ...
- 18.4k
5
votes
1
answer
385
views
Confusion with formally unramified = immersion and formally smooth = submersion
From this MO question I learned to tentatively think of formally unramified arrows as immersions and of formally smooth arrows as submersions.
I'm trying to semi-formally handwave myself into ...
- 10.3k
14
votes
2
answers
487
views
Which spaces have enough curves
Let $\mathbf{Top}$ be the category of topological spaces, and let $I\in\mathbf{Top}$ be the unit interval $I=[0,1]\subset\mathbb{R}$. For any space $X$, let $|X|$ denote the underlying set of points; ...
- 8,327
2
votes
0
answers
60
views
Dual equivalence for multioperators
This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
- 997