# Tagged Questions

**10**

votes

**1**answer

642 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 ...

**25**

votes

**9**answers

1k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**18**

votes

**0**answers

555 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 ...

**6**

votes

**3**answers

404 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 ...

**2**

votes

**0**answers

66 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 ...

**5**

votes

**4**answers

644 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 ...

**6**

votes

**0**answers

228 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 ...

**32**

votes

**2**answers

1k 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 ...

**7**

votes

**1**answer

298 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 $p \preceq r$ ...

**10**

votes

**3**answers

390 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?

**8**

votes

**6**answers

1k views

### Giving $Top(X,Y)$ an appropriate topology

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 $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...

**2**

votes

**2**answers

228 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 ...

**10**

votes

**1**answer

410 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 ...

**9**

votes

**1**answer

308 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 ...

**7**

votes

**2**answers

766 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 ...

**3**

votes

**1**answer

224 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.

**6**

votes

**2**answers

294 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 ...

**4**

votes

**0**answers

208 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 ...

**4**

votes

**3**answers

633 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 ...

**3**

votes

**1**answer

481 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 ...

**6**

votes

**2**answers

412 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, ...

**1**

vote

**1**answer

360 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 ...

**3**

votes

**1**answer

336 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 ...

**1**

vote

**0**answers

316 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 ...

**7**

votes

**2**answers

258 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 ...

**2**

votes

**1**answer

241 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 ...

**1**

vote

**0**answers

132 views

### Local cartesian closedness in the category of compactly generated spaces

According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...

**45**

votes

**28**answers

4k 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, ...

**1**

vote

**0**answers

309 views

### Direct Limits and Limits of Nets

A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...

**5**

votes

**2**answers

433 views

### A subcategory of top where subspaces and subobjects coincide?

I think that my question is easily answerable. The question is: What is a nice subcategory of topological spaces where the subobjects are subspaces. I would like the category of compactly generated ...

**2**

votes

**1**answer

226 views

### Induced pretopologies on sSet

Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...

**5**

votes

**2**answers

457 views

### Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...

**8**

votes

**3**answers

624 views

### Localic locales? Towards very pointless spaces by iterated internalization.

One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...

**7**

votes

**0**answers

234 views

### The self-duality of topological compactness

The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I toyed ...

**19**

votes

**3**answers

2k views

### Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...

**3**

votes

**4**answers

585 views

### On locally convex (and compactly generated) topological vector spaces

Part 1:
How big is the category $TVS_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)?
In other words (and less cheekily), is there a free locally convex TVS having any ...

**25**

votes

**3**answers

2k views

### Why are profinite topologies important?

I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...

**11**

votes

**1**answer

345 views

### Do strict pro-sets embed in locales?

It is well-known that the category of profinite groups (by which I mean Pro(FiniteGroups), i.e. the category of formal cofiltered limits of finite groups) is equivalent to a full subcategory of ...

**13**

votes

**5**answers

868 views

### What abstract nonsense is necessary to say the word “submersion”?

This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall ...

**6**

votes

**3**answers

2k views

### compact-open topology

Is there a natural reason for defining the compact-open topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another ...

**10**

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 ...

**1**

vote

**1**answer

661 views

### How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...

**3**

votes

**1**answer

384 views

### Functoriality of base change

Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism ...

**10**

votes

**1**answer

696 views

### Category Theory / Topology Question

Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly.
Let $\mathcal{C}$ ...

**3**

votes

**2**answers

276 views

### How do you know when a reflective subcategory of Top is quotient-reflective?

A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a reflective subcategory if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow ...

**16**

votes

**0**answers

480 views

### Is there a category of topological spaces such that open surjections admit local sections?

The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of ...

**13**

votes

**5**answers

549 views

### How can one characterise compactness-by-experiment?

There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that fit the ...

**5**

votes

**4**answers

2k views

### Why are inverse images more important than images in mathematics?

Why are inverse images of functions more central to mathematics than the image?
I have a sequence of related questions:
Why the fixation on continuous maps as opposed to open maps? (Is there an ...

**1**

vote

**1**answer

468 views

### When is the realization of a simplicial space compact ?

Suppose $X$ is a simplicial space of dimension $M$ (i.e. all simplices above dimension $M$ are degenerate). The claim is:
$|X|$ is compact. iff $X_n$ is compact for each $n$.
Suppose each $X_n$ is ...

**1**

vote

**0**answers

178 views

### Is the realization of a proper map of simplicial spaces proper ?

Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...