Tagged Questions
4
votes
1answer
65 views
Sheaf as a functor
Let $X$ be any topological space, $S$ - any category (e.g. of sets). Consider a new category $C$: its objects are only open subsets of $X$ and a set of morphisms from $U$ to $V$ is nonempty if and ...
2
votes
1answer
85 views
Quotient space and Retractions
I'm trying to learn something about topology and category theory.
Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia)
For an ...
4
votes
0answers
58 views
Products of sites
Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
0
votes
1answer
33 views
Paths between 0-cells in a classifying space. II
Let $\mathcal{C}$ be a small category and $X,Y$ objects within.
If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
37 views
Paths between 0-cells in a classifying space.
Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
2
votes
0answers
63 views
Pullbacks as manifolds versus ones as topological spaces
Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd).
Questions:
Suppose that we ...
3
votes
1answer
67 views
Properties of the Category of topological spaces with $n$ basepoints.
I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...
5
votes
1answer
80 views
A new(?) partial order on the set of continuous maps
Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
9
votes
1answer
167 views
Completion as a functor between topological rings
In the following all rings are assumed to be commutative and unitary.
Preliminaries:
For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
9
votes
2answers
197 views
Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?
Every metric space is associated with a topological space in a canonical way. According to this source, this amounts to a full functor from the category of metric spaces with continuous maps to the ...
7
votes
0answers
127 views
What properties are preserved under a measurable mapping?
Although in an abstract category the morphisms are not explicitly defined, in a concrete example (model theory?), morphisms are (always/usually?) mappings that preserve some properties.
In the ...
2
votes
1answer
67 views
How to apply product object in category to get product sigma algebra/topology/set systems?
Following is similar to my earlier questions, but try to understand them from category theory. Mariano said it was possible in a comment, but I don't know how.
An object $X$ is the product of a ...
4
votes
1answer
159 views
When does it make sense to define a base of a set system?
In a topology, a base is defined to be a class of subsets such that
every open set is the union of some members of it.
In a convexity
structure, a base is defined to be a class of subsets ...
3
votes
1answer
118 views
Do pushouts preserve regular monomorphisms?
In $\mathbf{Top}$, if $A$ is a subspace of $X$ and $f: A \to Y$ is a continuous mapping, then $Y$ embeds into $X \cup_f Y$. I wonder if this generalizes to an arbitrary category.
Consider the pushout ...
5
votes
1answer
92 views
Pushout of open map is open
I have been struggling with the following problem.
Consider the pushout for topological spaces (or adjunction space) $B \cup_A C$ obtained by gluing together $B$ and $C$ along $A$ by means of ...
5
votes
1answer
144 views
When and why do products preserve pushouts?
Let $A,B,C$
topological spaces and then $D$
the pushout of a diagram
$$B\stackrel{b}{\leftarrow}A\stackrel{c}{\rightarrow}C.$$
It seems logical to me that for a fifth topological space $E$ the ...
7
votes
1answer
88 views
pseudo-inverse to the operation of turning a metric space into a topological space
The construction of turning a metric space $(X,d)$ into a topological space by inducing the topology generated by the open balls gives rise to a functor $Met\to Top$ for any reasonable category $Met$ ...
1
vote
1answer
81 views
Pullback in $\mathrm{TOP}$
The pushout in the category of topological spaces is given by the gluing of spaces along continuous maps. Does there exist a similar "easy" topological description of the pullback? Does it even always ...
8
votes
3answers
278 views
What are the epimorphisms in the category of Hausdorff spaces?
It appears to be the case that the epimorphisms in $\text{Haus}$ are precisely the maps with dense image. This is claimed in various places, but a comment on my blog has made me doubt the source I got ...
3
votes
0answers
61 views
What is special about simplices, circles, paths and cubes?
There are some ubiquitous families of graphs — the complete graphs (or simplices) $K_n$, the circle graphs $C_n$, the path graphs $P_n$, and the hypercube graphs $Q_n$ — that intuitively ...
11
votes
0answers
199 views
Understanding Alexandroff compactification
Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
0
votes
1answer
83 views
How products in Top and Set are related?
Are product morphisms for a categorical product in Top the same as for categorical product morphisms in Set?
More generally: How product morphisms for Top are characterized?
0
votes
0answers
101 views
Why Top is not cartesian closed? [duplicate]
Possible Duplicate:
Is Top provably not cartesian closed?
I have heard that Top is not cartesian closed.
Why Top is not cartesian closed? Which criteria of being cartesian closed is ...
1
vote
0answers
52 views
Adjoint to the Hom functor in Boolean rigs
What I wanna ask is about analogies to the tensor product for commutative boolean rings.
What I mean by commutative boolean ring is set with two operations, + and *, and two identities, 0 and 1, as is ...
-3
votes
1answer
87 views
Does an isomorphism induce an order isomorphism?
Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a
\curlyvee b$ if only if there is a non-least element $c$ such that $c
\leqslant a \wedge c \leqslant b$.
Let ...
1
vote
0answers
114 views
Applications of monads in general topology?
What are applications of monads in general topology?
For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
1
vote
1answer
143 views
Formal Definition/counter part in mathematics for “Objects” of Object Oriented Models [closed]
I'm a newbie in both formal mathematics and theoretical computer science, so please bear with me if you find my question is not properly framed.
Object Oriented Modeling seems very useful in defining ...
4
votes
1answer
148 views
Universal property of initial topology
I'm learning some category theory and I thought I had understood universal objects but maybe I have not because I cannot write down the definition of initial topology in terms of categories.
My ...
3
votes
1answer
81 views
Categorification of the (co-)induced topology
In second semester analysis we learned about the product topology which is quite easy to categorify using limits. However, we also learned about the coinduced topology $\mathfrak{V}$ induced by $f: X ...
2
votes
3answers
129 views
Are groups algebras over an operad?
I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In ...
7
votes
0answers
141 views
Is there an abelian cat of topological groups?
There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So i'm wondering:
Is there a "suitably well ...
3
votes
2answers
191 views
Clopen subsets of the Cantor set.
In our Topology course, we have been studying the anti-equivalance of categories between zero-dimensional compact Hausdorff spaces and Boolean algebras. The Cantor set has come up a lot, and I have a ...
3
votes
1answer
304 views
Meaning of “a mapping preserves structures/properties”
Sometimes I see something like "a mapping preserves the structures
of its domain and of its codomain". From Wiki about morphisms in category theory:
a morphism is an abstraction derived from ...
11
votes
0answers
247 views
Are limits categorical limits? [duplicate]
Possible Duplicate:
Category-theoretic limit related to topological limit?
I was wondering if there exists some way to re-interpret "analytical" limits
$$ \lim_{x\to c}\; f(x)$$
as ...
2
votes
1answer
121 views
Existence of a structure-preserving mapping between two spaces?
I have some questions, but not sure if they are meaningful:
Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there
exist a measurable mapping from $X$ to $Y$?
Suppose $X$ and $Y$ are ...
3
votes
1answer
93 views
Representability of strange functor
Consider the category Top of topological spaces. Consider the contravariant functor from Top to Set sending a topological space X to the set of all opens of X. Is this functor representable?
What if ...
7
votes
2answers
362 views
Category-theoretic limit related to topological limit?
Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...?
To be more precise, is there a ...
2
votes
1answer
57 views
Characterization of “Smallness” of the category of coverings over a topological space
Fixed a connected topological space $X$ it's an exercise to show that, if $X$ admits a universal covering $Y \rightarrow X$, then the category $C$ of finite covering spaces of $X$ is small.
I'm ...
4
votes
1answer
181 views
Alternative name for “closed set”
It is usually argued (and also joked about) that classifying sets into open and closed is a bit paradoxical, since sets can be open and closed at the same time, or neither. This can be analyzed very ...
5
votes
1answer
166 views
Adjunction space is a pushout
I would like to show that the diagram
$$\begin{array}{}
A & \stackrel{f}{\longrightarrow} & Y \\
i \downarrow & & \downarrow {\phi_2} \\
X & ...
