Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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333 views

Metric vs metrizable spaces

A. Helemskii in the book "Lectures on functional analysis" write (in my horrible translation): The category of Hausdorff topological spaces (morphisms are continuous maps) contain the full ...
3
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1answer
546 views

Free crossed modules

A crossed module (over groups) $\mathcal{M} = (H,G,\partial)$ is a homomorphism $\partial\colon H \to G$ (called the boundary) together with an action $\alpha\colon (g,h) \mapsto {}^gh$ of $G$ on $H$ ...
3
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2answers
154 views

Concrete balanced category

A category is called balanced if every bimorphism is an isomorphism. Consider a concrete category such that every bijective morphism is a isomorphism. Does the category is balanced? Does converse is ...
2
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1answer
97 views

Commutativity of diagram involving two arrows

Hi suppose I have a diagram that looks like this: but where we only have $fe = hf'$ and $ge = hf'$. What would I call the square? I can't say that it commutes yes? Is it true that in general given ...
0
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1answer
91 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
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100 views

In topology $X$ is also $Y$ means homeomorphic?

E.g. $\mathbb{R}P^n$ is also the quotient space $S^n / (v \sim -v)$. And when is it safe to refer to a space as one of it's homeomorphic spaces and perform further deductions from that homeomorphic ...
3
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1answer
98 views

Definition of equalizer for $\textbf{Sh}(X)$

Let $\textbf{Sh}(X)$ denote the category of all (set - valued) sheaves on a topological space $X$. My question is: Given sheaves $F,G \in \textbf{Sh}(X)$ and morphisms $\varphi : F \to G$ ...
2
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1answer
134 views

Understanding Basic Categorical Duality with an Example from Group Theory

I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation. Let $H$ be any nontrivial subgroup of the alternating ...
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3answers
186 views

Mac Lane exercise - Elegant comma category exercise proven by S.A Huq

I came across this question whilst reading Mac Lane's "Category Theory for the working mathematician" and it struck me as quite a clever idea: "Given parallel functors $T,S: D \rightarrow C$, show ...
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2answers
103 views

Study of category and functor

I am going to start study of Category and Functors and sheaf theory. What are good text/ lecture notes to start with? I have not done any prior course on category theory and homological algebra or ...
11
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3answers
340 views

categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
0
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1answer
68 views

Product between a functor and a distributor

Given a distributor $\phi\colon \mathbf{A}\not\rightarrow \bf B$ and a functor $F\colon \bf B\to X$ I can define $F\otimes\phi$ to be the functor $\bf A\to X$ given by ...
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1answer
346 views

Pushout in Set example 2.6.2.2. in David Spivak's Book

Trying to understand Example 2.6.2.2. on page 50 of David I. Spivak's book "Category Theory for Scientists," Spivak gives $X$ as the interval $[0..1]=\{x\in\mathbb{R}|\,0\le x\le 1\}$, $Y$ as the ...
5
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2answers
175 views

What is a noetherian category?

What is a noetherian category? I'm a little bit familiar with category theory, but I've no idea what this could be. Do you know what it is good for or examples?
8
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1answer
392 views

Examples of mathematical statements made with adjoint functors

I am wondering if it is possible to use the adjoint functors in topos theory for statements in analysis. Any examples would be warmly welcomed. Though I would prefer simpler, atomic, lemmas or ...
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0answers
111 views

Can curvature be defined in Topos Theory?

I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context. Topos theory claims that it is a geometric ...
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7answers
1k views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
8
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1answer
158 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
2
votes
1answer
114 views

pullback square of factor groups

Let H and K be normal subgroup of a group G. The following square is always a pullback square? $$\begin {matrix} G/H\cap K\rightarrow &G/K\\ \downarrow&\downarrow\\ G/H\rightarrow&G/HK\\ ...
5
votes
1answer
289 views

Are there dual logic gates?

In category theory the usefulness of dualising such devices as monoids to comonoids has been shown, where multiplication which takes two inputs to one output is dualised to comultiplication which ...
8
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1answer
505 views

Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?

Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
7
votes
1answer
112 views

Under what choice assumptions is there a monoid structure on every set?

The question arose when discussing possible cardinalities of hom-sets of whether it's any weaker than the axiom of choice that there exists a monoid of every cardinality. It's well known, or at least ...
7
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1answer
91 views

Characterizing categories by size

Usually one distinguishes five classes of categories by size, and there are examples for all of them: finite categories locally finite categories small categories locally small categories large ...
5
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3answers
238 views

Applications of Category Theory Outside of Mathematics (the discipline)

I apologize if this is off-topic, but I wish to know what applications have other disciplines made of category theory. I have heard that linguistics, computer science, and philosophy all make use of ...
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1answer
105 views

Understanding morphism of category

Please help me understand 'arrows' (morphism) in Category theory. For a Category A let natural numbers be 'objects' and let's assume that I want to define summation (+) as the composition then what ...
13
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1answer
205 views

A category where maps are factorizations - what is this called?

Let $\mathcal C$ be a category, and define $\mathcal D$ to be the category whose objects are maps in $\mathcal C$, and where a map $f\to g$ is a factorization $pfq=g$. Composition of $(p_1,q_1):f\to ...
33
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1answer
943 views

Does “cheap nonstandard analysis” take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the ...
4
votes
2answers
78 views

Identity of Initial Elements in a Category

I don't know much (really, anything) about Category Theory or Structural Set Theory, but I happened across this blog post which piqued my interest. The author claims: But in Structural Set ...
4
votes
1answer
195 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
2
votes
1answer
339 views

Definition of sheaf using equalizer

Wikipedia give sheaf property using equalizer diagram by saying sheaf property means for any open cover $\{U_i\}$ of $U$ $$F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop ...
5
votes
1answer
271 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
2
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1answer
82 views

How to prove that initial arrows in Haus coincide with topological embeddings?

In Joy of Cats it is stated that in category $\textbf{Haus}$ initial arrows coincide with topological embeddings (pg 135). This can be proved by showing that initial arrows in $\textbf{Haus}$ are ...
5
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101 views

domain of initial $f : X \rightarrow Y$ in Haus equipped with coarsest topology?

If $f:X\rightarrow Y$ is initial in category Top then it is easy to proof that (!) the topology on $X$ is the set of preimages of open sets in $Y$. Just construct topology $Z$ having the same ...
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1answer
144 views

Systematizing graph morphisms

Trying to systematize possible notions of graph morphisms I came about the following classification: A morphism $f$ which sends a graph $G$ to another graph $G'$ is – first of all – ...
3
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0answers
85 views

Proof in Hungerford about natural isomorphisms of multifunctors

Reading Hungerford's Algebra I encountered the following exercise: (A) Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Let $S,T: \mathcal{C} \to \mathcal{D}$ be covariant functors. If $\alpha: ...
2
votes
2answers
138 views

Is taking cokernels coproduct-preserving?

Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms. Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the ...
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0answers
38 views

Possible number of endofunctors

The discrete category with countably many objects and morphisms has uncountably many endofunctors (= the number of functions from $\mathbb{N}$ to $\mathbb{N}$). Which categories with countably many ...
2
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3answers
214 views

Intuitive Explanation of Hom Functor Property

I am new to category theory and keep running across it while studying algebra. Whenever I see the $\mathrm{Hom}$ functor mentioned (in the context of modules), two of its basic properties are listed: ...
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0answers
34 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
13
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3answers
320 views

Is there a suitable definition in categories for a closed continuous function in topology?

Working in the category of topological spaces is it possible to give a 'categorical' definition for 'a closed continuous function'? I mean something like: 'a closed continuous function' is an arrow in ...
9
votes
1answer
281 views

What distinguishes topological spaces from graphs?

Topology would not "work" if one reverted the "direction" in the definition of continuous maps $f$: $$\text{open}(x) \rightarrow \text{open}(f(x))$$ It has to be $$\text{open}(f(x)) \rightarrow ...
8
votes
1answer
116 views

Universal properties of “interesting” families of graphs

Can cycles – and/or other "interesting" families of graphs like paths, trees, hypercubes, etc. – be characterized by a universal property in the category of graphs? I admit that there is ...
1
vote
1answer
47 views

Family of maps from inverse system to another space

The universal property of an inverse limit $\lim_{\leftarrow} X_\alpha$ allows one to define from a compatible system of maps $\psi_\alpha: Y \rightarrow X_\alpha$ a unique map $\psi: Y \rightarrow ...
10
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4answers
356 views

What precisely is lost when considering proper classes rather than sets?

Motivated by concerns over the foundational issues vis-a-vis category theory. What is the essential useful characteristic of sets that is lost when instead considering proper classes? Referring to ...
4
votes
2answers
200 views

An example of a monomorphism that is not an equalizer in “Abstract and Concrete Categories — The Joy of Cats”

I am sorry, maybe i am just confused or have not understood some definition, but i do not understand the following remark in Abstract and Concrete Categories -- The Joy of Cats on page 117: ...
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2answers
238 views

Completion of a metric space in categorical terms

Is it possible to define the completion of a metric space using categorical terms?
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1answer
255 views

Whats the diffrence between Products and Coproducts

So I just started in on Category theory (reading the quintessential text, "Categories for the Working Mathematician"), and I am trying to get my head around the difference between Products and ...
7
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1answer
226 views

Hartshorne's weird definition of right derived functors and prop. III 2.6

There is something very weird with the way Hartshorne defines right derived functors. Hartshorne p 204 Let $\mathfrak A$ be an abelian category with enough injectives, and let $F \colon \mathfrak ...
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1answer
61 views

Different direct product in a category and its full subcategory

A question related to Continuing direct product on a subcategory. Let $F$ is a full subcategory of a category $G$. I denote $\operatorname{Ob}X$ the set of objects of a category $X$. Is it possible ...
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1answer
28 views

Continuing direct product on a subcategory

Let $F$ is a full subcategory of a category $G$, both categories having binary direct product. Is it always true that there is such a binary direct product in $G$ that it is a continuation of a ...