All Questions
29
questions
6
votes
1
answer
430
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 ...
- 59k
4
votes
0
answers
192
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 \...
- 123
8
votes
0
answers
159
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{...
- 14.8k
9
votes
1
answer
396
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 ...
- 1,803
7
votes
1
answer
321
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 ...
- 193
9
votes
0
answers
185
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 $Y$,...
- 14.8k
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 ...
- 8,027
2
votes
0
answers
193
views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is ...
- 12.4k
34
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 6,757
4
votes
0
answers
396
views
Brouwer's fixed point theorem and the one-point topology [closed]
I posted this question last week on Math SE and got upvotes and helpful comments that allowed me to make the question more precise https://math.stackexchange.com/q/3765546/810513. As I did not get an ...
- 205
3
votes
0
answers
131
views
Colimits of weak Hausdorff $k$-spaces
Notations:
$\mathbf{T}$ is the category of weak Hausdorf $k$-spaces.
$\mathbf{K}$ is the category of $k$-spaces.
Fact: The inclusion functor $\mathbf{T} \subset \mathbf{K}$ is a right adjoint. It ...
- 3,697
1
vote
0
answers
194
views
Surjectivity of colimit maps for topological spaces
From this post and to (co)completness of the category Top of topological spaces and continuous functions we know that for any diagram $B_i$ and an object $A$ in Top, there are natural maps of sets:$\...
- 5,001
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
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
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
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
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
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
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
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 ...
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
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
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
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
11
votes
9
answers
1k
views
Proving the impossibility of an embedding of categories
A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...
- 5,569
3
votes
1
answer
356
views
Is the coproduct of fibrant spectra fibrant again?
Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...
- 51