Various structures are studied in category theory using properties of objects and morphisms between them. Many construction 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 ...
2
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1answer
98 views
Understanding the Yoneda lemma
I'm having difficulty understanding the Yoneda lemma. In particular, the proof isn't that obvious to me. Please, could someone explain to me the error of my current understanding..
The Yoneda lemma ...
3
votes
2answers
56 views
What is the point of extremal epimorphisms in category theory? Why not just use strong epis instead?
I've been trying to get my head around the various types of epimorphisms you get in category theory, but I can't see why anyone uses "extremal" epis as opposed to the slightly less general notion of ...
7
votes
4answers
222 views
Why is 'isomorphism' defined more generally in Category theory than in Abstract Algebra?
I read in Awodey's Category Theory book that the definition of isomorphism in category theory is more general than the one in abstract algebra. For example, he says, the definition of isomorphism from ...
5
votes
0answers
61 views
Limits and colimits in the category of fields
It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...
5
votes
1answer
100 views
Elephant: how do I prove Lemma 2.1.7, section C2.1?
I'm referring (also for notations and terminology) to P. Johnstone, Sketches of an Elephant. A Topos Theory Compendium. Volume I. Clarendon Press. Oxford, 2002. The Lemma can be found at page 540.
I ...
2
votes
2answers
131 views
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
votes
2answers
32 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 ...
9
votes
1answer
115 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 ...
2
votes
2answers
56 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 combinatorial ...
3
votes
0answers
42 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 ...
1
vote
1answer
103 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 ...
2
votes
1answer
69 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 ...
6
votes
0answers
67 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
votes
1answer
62 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...
...
0
votes
1answer
56 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 ...
-1
votes
1answer
66 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)}):= ...
2
votes
1answer
59 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 ...
7
votes
2answers
101 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 ...
2
votes
1answer
73 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 $2^{\beta}: 2^{B_1} \to 2^{B_2}$ ...
6
votes
2answers
92 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 $(G,*),$ ...
8
votes
1answer
138 views
Stone's Representation Theorem and The Compactness Theorem
If you're working on $ZF$ 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 Boolean algebras ...
1
vote
1answer
27 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?
1
vote
1answer
53 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 ...
4
votes
3answers
150 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 ...
5
votes
1answer
67 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 ...
16
votes
1answer
487 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
votes
1answer
56 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. ...
5
votes
1answer
104 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 ...
3
votes
1answer
77 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 ...
10
votes
2answers
219 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 ...
0
votes
1answer
50 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 ...
1
vote
1answer
58 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?
2
votes
1answer
60 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 ...
0
votes
1answer
31 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 ...
6
votes
2answers
84 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?
6
votes
2answers
45 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?
1
vote
1answer
39 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 ...
0
votes
1answer
41 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
votes
1answer
84 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
votes
1answer
45 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
36 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 ...
1
vote
1answer
38 views
Morphisms in the category of group presentations
What are the morphisms in the category of group presentations?
3
votes
1answer
47 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
50 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 ...
4
votes
1answer
68 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
votes
0answers
33 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
votes
1answer
55 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 ...
7
votes
1answer
78 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 ...
3
votes
1answer
55 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?
3
votes
1answer
51 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 ...
