1
vote
1answer
91 views

Foundations book using category theory?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
9
votes
3answers
118 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
10
votes
0answers
258 views

Is this $really$ a categorical approach to $integration$?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
1
vote
0answers
34 views

Categorization of Mono, Epis in a Category

Let $C$ be a category. Then the following implications on variants for monos hold: Iso $\implies$ SplitMono $\implies$ RegMono $\implies$ StrongMono $\implies$ ExtMono $\implies$ Mono, And dually ...
1
vote
1answer
54 views

Is there a way to phrase “there does not exist a universal set” in structural language?

Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy ...
2
votes
2answers
75 views

Algebraic Object in Topology

Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making ...
4
votes
2answers
102 views

Size Issues in Category Theory

Barr and Wells state in their text Toposes, Triples and Theories (pdf link) It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. ...
6
votes
3answers
171 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
1
vote
0answers
38 views

Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
5
votes
2answers
349 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
10
votes
3answers
172 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
2
votes
2answers
67 views

Can things go wrong if we declare objects to be arrows?

I am familiar with categories and also with 'categories without objects'. In fact a category is completely determined by its set of arrows and the objects can be missed. Nevertheless the objects are ...
4
votes
0answers
81 views

Earliest precursor to category theory

In the Historical notes section of the Wikipedia article on category theory, it is mentioned that in 1942-1945, Samuel Eilenberg and Saunders Mac Lane, in the course of their work in algebraic ...
2
votes
0answers
177 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
2
votes
2answers
66 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
8
votes
1answer
66 views

Why an integral symbol for the category of elements of a presheaf?

Let $\mathbf C$ be a category and $P \colon \mathbf C^{\rm op} \to \mathbf{Set}$ a presheaf. One can associate to $P$ the category of elements of $P$ (also called Grothendieck construction over $P$), ...
5
votes
3answers
773 views

What does “Arrows are more important than objects” really mean?

I am a final year undergraduate student and I am trying to learn category theory. I am familiar with the basic notions. I am reading Pareigis's notes, ...
10
votes
4answers
146 views

Why are particular combinations of algebraic properties “better” (richer and more pervasive) than others?

I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not ...
8
votes
2answers
135 views

How to introduce category theory to a high school audience?

I am a mathematician with background in Category Theory. I have been asked to give a 20 minute talk about my area of research to an audience of talented high school students and school mathematicians ...
4
votes
1answer
68 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
2
votes
1answer
68 views

If $X$ and $Y$ are objects of $\mathrm{Set}$, is there any reason not to regard $\mathrm{Hom}(X,Y)$ as an object, too?

Suppose $X$ and $Y$ denote objects of $\mathrm{Set}$. Is there any reason not to say: therefore, $\mathrm{Hom}(X,Y)$ is itself an object of $\mathrm{Set}$? Furthermore, is there any reason not to ...
10
votes
1answer
240 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
4
votes
1answer
70 views

Is there a specific notion as to what 'forgetfulness' is?

Concrete categories $A$ carry a forgetful functor $U:A\rightarrow Set$, whose left adjoint if it exists is the free functor. There are other forgetful functors such as $U':Ass\rightarrow Vect$ whose ...
6
votes
1answer
68 views

How to look at a polynomial ring based on a ring that is not commutative?

When I first met polynomial rings $R[X]$ I wondered: 'where do they come from?' Later the idea that - if $R$ is commutative - they could be interpreted as $R$-algebras free over a singleton brought ...
6
votes
2answers
82 views

How should we think of maps to the intial object?

A final object in a category is one that has a unique map from any other object. An intuitive way of thinking about the final object is as the 'point'. Then we think of maps from the 'point' to ...
6
votes
1answer
76 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
0
votes
2answers
69 views

Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$ X\times Y \to X \quad\text{ and }\quad X\times Y \to Y. $$ Usually I denote them by $\pi_X$ and ...
3
votes
2answers
70 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
2
votes
2answers
112 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
6
votes
2answers
181 views

What is a (the?) good starting point for learning the modern “higher” mathematics?

As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself ...
7
votes
2answers
192 views

How much Category theory one must learn?

I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should every Mathematics student who is not planning to specialize ...
7
votes
1answer
126 views

Is there any point in a logician studying $\infty$-categories?

My primary areas of interest lately have been set theory, logic, and category theory, so naturally topos theory has been a large part of what I'm learning (in between getting caught up on some other ...
2
votes
1answer
30 views

A case where we have a functor, and we are looking for the right adjoint

Most of the examples of adjoint functors I saw ''in the wild'' have a right adjoint forgetting a part of structure, and left adjoint recovering it in the most efficient/general way. Often, a functor ...
5
votes
1answer
426 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
0
votes
1answer
60 views

ENS is an abbreviation of?… [duplicate]

In CWM Mac Lane uses the term $\mathbf {ENS}$ for a category having as objects the subsets of a given set and as morphisms the functions from these sets to these sets. What is abbreviated by the ...
6
votes
1answer
135 views

Is this an accurate portrayal of what category theorists with an interest in the foundations are actually trying to achieve?

The problems faced by the foundationalist programmes of the last century included trying to decide: Which is the one true logic on which all mathematics should be based? See also, Brouwer-Hilbert ...
3
votes
2answers
223 views

Does the adjugant of additive functors between abelian categories preserve the abelian structure of the hom-set?

I think the following is a counter-example. I noticed it when trying to prove that the sheafification functor induces isomorphism on the stalks (Vakil 2.4M). As in Vakil (2.6.3) the stalk functor is ...
2
votes
2answers
120 views

Category of natural numbers with divisbility?

It occurred to me earlier today that one can form a category by taking the objects to be the positive natural numbers and postulating a morphisms from $a$ to $b$ iff $a$ divides $b$. (That is, there ...
2
votes
2answers
157 views

Category for measure spaces?

I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems ...
1
vote
2answers
141 views

What is a real-valued random variable?

This question arose when someone (and surely not the least!) commented that something like $\left(X\mid Y=y\right)$ , i.e. $X$ under condition $Y=y$, where $X$ and $Y$ are real-valued random variables ...
8
votes
1answer
97 views

Encyclopedia of categories

Is there a digest of the most widely used categories with their basic properties mentioned? So I could, say, learn from it in a couple of minutes that the category with totally ordered sets as objects ...
9
votes
1answer
336 views

On a joke of Yoneda embedding

I have heard a joke like this: The Yoda embedding, contravariant it is. And a joke concerning "How to put an elephant into a refrigerator", a comment from "Category Theorist" says Isn’t this ...
10
votes
3answers
290 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 ...
28
votes
7answers
884 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
votes
7answers
341 views

Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft ...
10
votes
2answers
234 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", ...
7
votes
1answer
123 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
72 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
140 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 ...
5
votes
4answers
243 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 ...