10
votes
1answer
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
9answers
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
0answers
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
3answers
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
0answers
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
4answers
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
0answers
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
2answers
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
1answer
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
3answers
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
6answers
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
0answers
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
3answers
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
28answers
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
0answers
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
2answers
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
1answer
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
2answers
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
3answers
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
0answers
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
3answers
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
4answers
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
3answers
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
1answer
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
5answers
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
3answers
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
9answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
5answers
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
4answers
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
1answer
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
0answers
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 ...