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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Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics

As I understand that there are at least two fundamental limits of the development of the mathematics: 1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...
2
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2answers
45 views

In Sets, describe the equalizer of the functions $f \circ p_1$ ,$ f \circ p_2 : A \times A → B$ as a (binary) relation on $A$

I am trying to solve Awodey 3.5.6a: Consider the category of sets. Given a function $f : A \to B$, describe the equalizer of the functions $f \circ p_1$, $f \circ p_2 : A \times A \to B$ as a ...
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1answer
161 views

Universal property characterizing $\Bbb R$

Is it possible to characterize the field of real numbers in a natural way with the language of category theory? For example, $\Bbb Q$ is the initial object in the category of ordered fields and $\Bbb ...
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3answers
186 views

Learning Combinatorial Species.

I have been reading the book conceptual mathematics(first edition) and I'm also about halfway through Diestel's Graph theory (4th edition). I was wondering if I was able to start learning about ...
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0answers
48 views

Iterated endofunctors

Suppose $F : \mathbb{C} \rightarrow \mathbb{C}$, with the following constraints: $F^{n+1}(\mathbb{C})$ is a subcategory of $F^{n}(\mathbb{C})$ for all objects $X \in \mathbb{C}$ there exists a ...
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1answer
161 views

Category theory - what's the intuition behind diagrams?

I'm new to category theory, and I'm trying to understand diagrams. What's the connection between the pencil-and-paper diagrams that I draw in my workbook, and the technical definition that a diagram ...
3
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1answer
113 views

Does the direct image functor on sheaves reflect epimorphisms?

Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
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0answers
73 views

Accessible introduction to category theory from the point of view of preorders. [duplicate]

Are there books renowned for introducing category theory in a very accessible way? An emphasis on the point of view that categories generalize preorders would be especially appreciated. My goal is to ...
2
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1answer
78 views

How does $C$ small imply $Set^{C^{op}}$ locally small?

I read in some notebook that $C$ small implies $Set^{C^{op}}$ locally small, but I don't see what is the reasoning used, because the Yoneda lemma is not mentioned so that it is probably not needed... ...
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1answer
101 views

Subcategory of a category consisting of collection of objects

The following is Exercise 3.7 from Aluffi's Algebra: Chapter Zero (available here): A subcategory $C'$ of a category $C$ consists of a collection of objects of $C$, with morphisms ...
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114 views

Composition of Morphisms where C is a category

The following is Exercise 3.1 from Aluffi's Algebra: Chapter Zero: Let $C$ be a category. Consider a structure $C^{(op)}$ with $\newcommand{\Hom}{\operatorname{Hom}}$ $Obj(C^{(op)}):= ...
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1answer
115 views

How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?

I'm reading Awodey's Category Theory (1st ed) and at page 166 I did not found out the proof of one of the remarks (remark 8.4): If $C$ is locally small, then $\mathsf{Sets}^{C^{\text{op}}}$ needs ...
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3answers
205 views

The category Set seems more prominent/important than the category Rel. Why is this?

There's a lot of talk about Set, but less about Rel. As an outsider to category theory, this surprises me, because Rel seems "more closed." In particular, The converse of a function needn't be a ...
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1answer
84 views

How exactly does a map represent an operation?

My question is related to exercise 5.7 in Sets for Mathematics by Lawvere and Rosebrugh (p. 105). We are given a map $\beta: B_2 \to B_1$ and a contravariant functor $2^{\beta}: 2^{B_1} \to 2^{B_2}$ ...
5
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2answers
138 views

Is a topological space a structure?

In model theory, a structure (or "model") is typically defined as a set together with some finitary relations and/or operations on that set. For instance, a group can be viewed as a pair $(G,*),$ ...
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1answer
212 views

Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
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1answer
50 views

Image of the composition of a kernel with a cokernel.

Let $ h:H\to G $ and $ k:K\to G $ be two normal monomorphisms and let $ f:H\ast K\to G $ theire coproduct. It is always true that $ h\text {coker} k $ and $ f\text {coker} k $ has the same image?
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1answer
297 views

The definition of the quotient category in abelian category.

I want to understand the definition of morphisms in this category. My question is how can I construct directed sets and direct systems, and therefore understanding the colimite. Definition: Given a ...
6
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4answers
324 views

What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
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1answer
414 views

Understanding Hom functions

I am very new to category theory. Started learning about this Hom sets/functions. I read $\operatorname{Hom}(S,T)$ as set of all functions from $S$ to $T$ but somehow this is a overloaded definition ...
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1answer
717 views

Why should a combinatorialist know category theory?

I know almost nothing about category theory (I have just skimmed the first chapters of Aluffi's algebra book), reading this question got me thinking... why should someone mostly interested in ...
2
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2answers
122 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
7
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1answer
648 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
4
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1answer
144 views

Mono's and Epi's in the category Rel?

Sorry to ask such a trivial question, but I can't find the answer anywhere. Question. What are the monomorphisms/epimorphisms in Rel? Furthermore, what's the standard terminology for describing ...
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2answers
355 views

coequalizers+pullbacks implies equalizers?

The question is on the title, I would like a hint on this exercise. This is what I've tried so far: Suppose we're given $f,g:A\rightarrow B$, let $h=\operatorname{Coeq}(f,g)$, then we have parallel ...
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1answer
165 views

Localization and initial objects

Let $A$ be a ring and let $S$ be a multiplicative subset of $A$. Why is the map from $A$ to $S^{-1}A$ initial among all $A$-algebras $B$? Why does localization not have to commute with respect to ...
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1answer
131 views

Direct limit of group

When we study shaves we have that a germ is the direct limit of groups (set, vector spaces). But how can I show that the direct limit of groups is a group?
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1answer
169 views

Can boolean homomorphisms of boolean algebras correspond to ultrafilters?

I am trying to solve 5th problem in Exercises 2.9 in Awodey's book on page 55: Show that for any boolean algebra $B$, boolean homomorphisms $h : B \to 2$ correspond exactly to ultrafilters in $B$. I ...
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1answer
40 views

$F:C\to D, G:D\to E$ are functors, $G$ has a right adj, $F$ is fully faithful, $G$ is faithful, $F$ is “relatively dense”. Does $F$ have an adj?

This is related to Questions 346458 and 348459. Suppose that $F:C\to D$ and $G:D\to E$ are functors such that $G$ has a right [left] adjoint $H$, $F$ is fully faithful, $G$ is faithful, and for each ...
7
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2answers
129 views

Is there a way to axiomatize the category of sets and relations?

The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?
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2answers
68 views

If $F:C\to D$ and $G:D\to E$ are functors, and both $GF$ and $G$ have a right adjoint, does $F$ has a right adjoint too?

Suppose that $F:C\to D$ and $G:D\to E$ are functors such that both of $GF$ and $G$ have a right adjoint. Is it true that also $F$ has a right adjoint? And what if only $GF$ has a right adjoint?
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1answer
48 views

Quick clarification: the pullback of the “multiply by 2” map

I've come across the following statement: The pullback of the "multiply by 2" map $\mathbb{Z}\to\mathbb{Z}$ along the "inclusion of zero" map $\ast\to\mathbb{Z}$ is the set of even integers. I'm ...
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1answer
60 views

Why the morphisms of the categories of monos of a category are pullbacks?

Let $C$ be a category and $Mono\left( C\right)$ the category which has: $Ob \left(mono\left(C\right)\right)=\left\{u: ux_1=ux_2 \implies x_1 =x_2\right\}$ ...
4
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1answer
90 views

Recovering an object from its category

Consider the category of groups (but the question arises for any category of mathematical object, basically). It is easy to read off what the automorphism group of a group is or what its cardinality ...
2
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1answer
54 views

How to find a direct product in a category?

Having a category how to find a direct product in this category? Is it entirely a guesswork (try this function, try that one) or is there a method for this?
2
votes
1answer
62 views

Is a functor category of an $\mathbf{Ab}$-category an $\mathbf{Ab}$-category itself?

In Weibel's An introduction to homological algebra, exercise 2.6.4 reads Show that $\operatorname{colim}$ is left adjoint to $\Delta$. Conclude that $\operatorname{colim}$ is a right exact ...
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1answer
49 views

Morphisms in the category of group presentations

What are the morphisms in the category of group presentations?
3
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1answer
86 views

$F:C\to D$, $G:D\to E$, $G$ has an adjoint, $F$ is fully faithful and for each $Z$ there is $X$ s.t. $F(X) = H(G(Z))$: Does $F$ has an adjoint?

Suppose $F:C\to D$ and $G:D\to E$ are functors. Assume that $F$ is fully faithful, $G$ has a left adjoint $H:E\to D$, and for each $Z \in E$ there exists $X \in C$ such that $F(X) = H(Z)$. Does $F$ ...
0
votes
1answer
68 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
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1answer
253 views

Is it possible to define a ring as a category?

Is it possible to define a ring as a category? For example, a group can be defined as a category with just one objet and all morphisms being iso.
2
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0answers
53 views

Categories of sheaves of finite homological dimension revisited

This is the question When is the category of (quasi-coherent) sheaves of finite homological dimension? revisited. I made some mistakes: I posted the question before I sign in, I tried to make it more ...
3
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1answer
86 views

When is the category of (quasi-coherent) sheaves of finite homological dimension?

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have ...
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1answer
96 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
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1answer
78 views

What is the cohomological explanation for the Condorcets voting paradox?

according to the nlab entry on the Condorcet Paradox in social choice (that is voting preferences may be circular even if voters preferences are not) has a cohomological explanation - what is it?
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1answer
77 views

Subcategory of Isomorphisms

There is a functor $\mathit{Iso} : \mathbf{Cat} \rightarrow \mathbf{Cat}$ which identifies the subcategory of a category in which only the isomorphisms appear as arrows — i.e. it strips off any ...
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2answers
541 views

Why is an empty set not a terminal object in categories $\mathsf{Top}$ and $\mathsf{Sets}$?

From Awodey: In any category $\mathsf{C}$, an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$, an object $1$ is terminal if for any object $C$ there is a unique ...
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3answers
96 views

Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$?

Let $F : J → C$ be a diagram of type $J$ in a category $C$. A cone to $F$ is an object $N$ of $C$ together with a family $ψ_X : N → F(X)$ of morphisms indexed by the objects $X$ of $J$, such ...
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3answers
252 views

Can morphisms in the category Set be partial functions?

In Set, objects are taken to be sets, with morphisms as functions. There are two questions, both are possibly related: Are the morphisms in Set required to be total functions, as opposed to partial ...
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1answer
44 views

$F:\bf C\to\bf D$ a functor with a right adjoint $G$ and $\bf S$ a full subcat of $\bf C$: When does the inclusion have a right adjoint?

Suppose a functor $F:\bf C\to\bf D$ has a right adjoint $G$, let $\bf S$ be a full subcategory of $\bf C$, and denote by $I$ the inclusion of $\bf S$ into $\bf C$. What are non-trivial assumptions ...
4
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1answer
200 views

Representable functors

Is the fact of being "representable" only defined for functors $\mathcal{C}\rightarrow\mathbf{Sets}$, or is there some similar concept for other kinds of functors? For example in an exercise sheet ...