# Tagged Questions

2answers
112 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", ...
3answers
75 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 ...
1answer
68 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 $F \dashv G$ ...
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 ...
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 ...
3answers
147 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 ...
1answer
482 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 ...
0answers
98 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 ...
3answers
289 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 ...
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 ...
3answers
153 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"?
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 ...
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 ...
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 ...
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 ...
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 ...
2answers
348 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 ...
2answers
196 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 ...
3answers
147 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 ...
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 ...
3answers
389 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, ...
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, ...
2answers
192 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 ...
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 ...
3answers
529 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 ...
1answer
336 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 ...
2answers
330 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; ...
1answer
514 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 ...
5answers
446 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 ...
3answers
515 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?
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' ...
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 ...
1answer
288 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 ...
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 ...
3answers
453 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 ...
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 $X$ ...
2answers
768 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 ...
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 ...
3answers
485 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 ...
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 ...
2answers
422 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 ...
3answers
982 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.
19answers
5k views

### Good book/lecture notes about category theory

what are the best books or lecture notes on category theory?
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