Tagged Questions
10
votes
2answers
117 views
Path Algebra for Categories
For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
0
votes
3answers
76 views
When is it important to distinguish between an object in a category and that object's identity morphism?
When is it important to distinguish between an object in a category and that object's identity morphism?
I am wondering if the only reason that we consider objects at all is to avoid infinitely ...
7
votes
1answer
69 views
Can we think of an adjunction as a homotopy equivalence of categories?
There is a way in which we can think about a natural transformation $\eta: F \rightarrow G$ as a homotopy between functors $F,G:\mathcal{C}\rightarrow \mathcal{D}$. Now, an adjunction $F \dashv G$ ...
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 ...
7
votes
2answers
103 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 ...
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 ...
16
votes
1answer
490 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 ...
8
votes
0answers
101 views
What are the “correct” modules over locally ringed spaces?
$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
9
votes
3answers
301 views
What is category theory useful for?
Okay so I understand what calculus, linear algebra, combinatorics and even topology try to answer, but why invent category theory? In wikipedia it says it is to formalize. As far as I can tell it sort ...
4
votes
2answers
57 views
Is there a “partial function” approach to subobjects in category theory?
Given a relation $f : X \rightarrow Y$, lets define that the source of $f$ is $X$, and that the domain of $f$ is the set of all $x$ such that there exists $y \in Y$ satisfying $(x,y) \in f$. Thus the ...
6
votes
3answers
154 views
In what sense is the forgetful functor $Ab \to Grp$ forgetful?
One sometimes hears about "the forgetful functor $Ab \to Grp$." Given that the image of an object under this functor is still abelian, in what sense is this "forgetful"?
2
votes
1answer
138 views
A pedantic question about defining new structures in a path-independent way.
Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of ...
5
votes
1answer
79 views
What is the minimum required background to understand articles in the nLab?
I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not ...
0
votes
3answers
147 views
Reference request - being rigorous about a common abuse of notation.
I've completely rewritten this question, in accordance with this advice.
As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
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 ...
1
vote
1answer
45 views
Interesting verification of functoriality
Functors and morphisms of functors (aka natural transformations) have become powerful tools in all areas of pure (and meanwhile also applied) mathematics. There are lots of nontrivial constructions of ...
15
votes
2answers
350 views
A structural proof that $ax=xa$ forms a monoid
During the discussion on this problem I found the following simple observation:
If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid.
This is trivial to prove by checking ...
5
votes
2answers
197 views
Learning category theory before abstract algebra
I'm reading this excellent pdf http://www.mimuw.edu.pl/~jarekw/pdf/Algebra0TextboookAluffi.pdf which is an algebra book, beginning with category theory and then use it for groups, rings,...
My ...
3
votes
3answers
148 views
How to do diagram chasing effectively?
I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3.
I have spent a lot of time ...
1
vote
2answers
106 views
The importance of parallel arrows in a commutative square
I noticed that whenever there is a commutative square, the relation it imposes on parallel morphisms is usually very important (e.g. natural transformations, pullbacks). In contrast, there's usually ...
10
votes
3answers
393 views
Category Theory usage in Algebraic Topology
First my question:
How much category theory should someone studying algebraic topology generally know?
Motivation: I am taking my first graduate course in algebraic topology next semester, and, ...
31
votes
6answers
1k views
Why don't analysts do category theory?
I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.
Recently, ...
1
vote
2answers
198 views
Why is full- & faithful- functor defined in terms of Set properties?
Wikipedia entry or Roman's "Lattices and Ordered Sets" p.286, or Bergman's General Algebra and Universal Constructions, p.177 and in fact every definition of full and/or faithful functor is defined in ...
4
votes
2answers
132 views
What things can be defined in terms of universal properties?
We can define some mathematical objects using universal properties, for example the tensor product, the free group over a set or the Stone–Čech compactification.
I'm wondering about how to develop my ...
30
votes
3answers
532 views
Why do we look at morphisms?
I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
17
votes
1answer
337 views
How to find exponential objects and subobject classifiers in a given category
In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject ...
6
votes
2answers
331 views
What should be the next step?
This is a soft/educational question and I'll flag it to be made community wiki.
A little bit of background, first. I am in my last undergraduate year, and I took a graduate course in category theory; ...
6
votes
1answer
520 views
Is category theory useful in higher level Analysis?
What I mean by higher level before this gets closed is functional analysis, complex analysis and harmonic analysis?
I've read looked at the examples in most category theory books and it normally has ...
7
votes
5answers
450 views
Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”
[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
6
votes
3answers
518 views
A Concrete Approach to Category Theory
Is there a way to learn Category Theory without learning so many concepts of which you have never seen examples?
5
votes
1answer
269 views
Mathematical structures
Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
6
votes
1answer
209 views
Concrete Categories Where Epis are Just Surjections
Before I begin let me provide some background to fix notation/make the post more readable to interested outsiders. In a category $\mathscr{C}$ we say that a morphism $X\xrightarrow{f}Y$ is an ...
5
votes
1answer
290 views
Mathematics needed for higher dimensional category theory?
I'm a undergrad(third year, Manchester uni) that is thinking of doing a PhD in this area or category theory in general.
Just wondering, what branches of Maths should I focus on? As I've been told ...
3
votes
1answer
225 views
Looking for an “arrows-only” intro to category theory
I have often seen it remarked in passing that the "collection of objects" that appears in the standard definition of a category is, strictly speaking, superfluous, and that it is possible to give an ...
10
votes
3answers
454 views
Looking for student's guide to diagram chasing
I'm teaching myself some category theory, and I find that I'm very slow with diagram chasing. It takes me some times a very long time to decide whether adding an arrow to a diagram preserves the ...
2
votes
3answers
104 views
Maps that assign points to maps
Consider a set $X$ and a set $Y$. Once can the define a map from $X$ to $Y$ that assigns to each point in $X$ a point in $Y$. On the other hand, if $F(X,Y)$ denotes the set of all functions from $X$ ...
11
votes
2answers
777 views
motivation and use for category theory?
From reading the answers to different questions on category theory, it seems that category theory is useful as a framework for thinking about mathematics. Also, from the book Algebra by Saunders Mac ...
6
votes
1answer
165 views
Introductory texts for weak $\omega$-categories
As I'm constantly running across higher categories these days, I'm wondering what is a good starting point to get into the theory?
While I am aware of nLab
and the n-Category Café, I am having a real ...
5
votes
3answers
490 views
Introduction to Bourbaki structures, and their relation to category theory
I just opened vol.1 of the Bourbaki treatise to take a look at how they define mathematical structure. I was amazed by its sheer complexity. Can you recommend an introductory text that wouldn't ...
5
votes
1answer
304 views
Reading commutative diagrams?
Sorry for this whole bunch of questions. Please note, that I know what a commutative diagram is, and that I can somehow read them, at least the simpler ones. But often enough the diagrams are labelled ...
9
votes
2answers
423 views
Category Theory with and without Objects
Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
6
votes
3answers
994 views
What are the prerequisites for learning category theory?
Is category theory worth learning for the sake of learning it? Can it be used in applied mathematics/probability? I am currently perusing Categories for the Working Mathematician by Mac Lane.
30
votes
19answers
5k views
Good book/lecture notes about category theory
what are the best books or lecture notes on category theory?
