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79 votes
5 answers
5k 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 ...
Qiaochu Yuan's user avatar
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 ...
Cayley-Hamilton's user avatar
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, ...
55 votes
3 answers
3k views

Duality between compactness and Hausdorffness

Consider a non-empty set $X$ and its complete lattice of topologies (see also this thread). The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
yada's user avatar
  • 1,691
42 votes
8 answers
5k views

What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...
Andrew Stacey's user avatar
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 ...
Adam P. Goucher's user avatar
38 votes
3 answers
5k 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 ...
Mike Shulman's user avatar
  • 64.1k
37 votes
13 answers
4k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful? I ...
Lehs's user avatar
  • 852
37 votes
5 answers
5k 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 ...
37 votes
1 answer
1k views

What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...
Nik Weaver's user avatar
  • 41.5k
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 ...
Tim Campion's user avatar
  • 59k
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:(...
Keshav Srinivasan's user avatar
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 ...
archipelago's user avatar
  • 2,954
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 ...
Claus's user avatar
  • 6,757
27 votes
3 answers
1k views

Possible categorical reformulation for the usual definition of compactness

Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...
Oleg Viro's user avatar
  • 373
24 votes
0 answers
899 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 $...
David Spivak's user avatar
  • 8,327
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 ...
coudy's user avatar
  • 18.4k
21 votes
2 answers
2k views

Colimits in the category of smooth manifolds

In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
Glen M Wilson's user avatar
21 votes
1 answer
813 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 $...
David Roberts's user avatar
  • 33.2k
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 ...
Ilan Barnea's user avatar
  • 1,324
17 votes
3 answers
1k 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 ...
Tom LaGatta's user avatar
  • 8,322
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 (...
Toby Bartels's user avatar
  • 2,584
17 votes
2 answers
549 views

In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?

For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
Gro-Tsen's user avatar
  • 28.7k
17 votes
0 answers
936 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 ...
Nicholas James's user avatar
16 votes
10 answers
3k views

References for homotopy colimit

(1) What are some good references for homotopy colimits? (2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
Kevin Walker's user avatar
  • 12.2k
16 votes
5 answers
4k 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 ...
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 ...
wonderich's user avatar
  • 10.3k
16 votes
5 answers
2k 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 ...
Theo Johnson-Freyd's user avatar
16 votes
1 answer
2k 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 ...
H. Shindoh's user avatar
16 votes
1 answer
552 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 ...
Mike Shulman's user avatar
  • 64.1k
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}\...
Amr's user avatar
  • 1,025
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 ...
Ivan Di Liberti's user avatar
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 ...
Alinas's user avatar
  • 181
15 votes
5 answers
670 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 ...
Andrew Stacey's user avatar
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), ...
Tim Campion's user avatar
  • 59k
14 votes
3 answers
688 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?
Gerrit Begher's user avatar
14 votes
3 answers
1k 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 $\...
Georg Lehner's user avatar
  • 1,823
14 votes
4 answers
1k 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 ...
David Feldman's user avatar
14 votes
3 answers
1k views

What is a monoidal metric space?

At time of writing, the highest rated answer to my question What is a metric space? is Tom Leinster's account of Lawvere's description of a metric space as an enriched category. This prompted my ...
Andrew Stacey's user avatar
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; ...
David Spivak's user avatar
  • 8,327
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 ...
Simon Henry's user avatar
  • 39.4k
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\...
user78499's user avatar
  • 141
13 votes
2 answers
732 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}...
Martin Brandenburg's user avatar
13 votes
1 answer
737 views

Is Top_4 (normal spaces) a reflective subcategory of Top_3 (regular spaces)?

I’m studying some category theory by reading Mac Lane linearly and solving exercises. In question 5.9.4 of the second edition, the reader is asked to construct left adjoints for each of the inclusion ...
user2734's user avatar
  • 1,371
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] \...
Valery Isaev's user avatar
  • 4,340
12 votes
4 answers
1k 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 ...
pre-kidney's user avatar
  • 1,279
12 votes
1 answer
2k 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 ...
M. Carmona's user avatar
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 ...
Daniel Miller's user avatar
11 votes
5 answers
1k views

Confusion over a point in basic category theory

"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
Cary's user avatar
  • 1,197
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'...
Kevin Arlin's user avatar
  • 2,859