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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5
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1answer
223 views

Adjunction space is a pushout

I would like to show that the diagram $$\begin{array}{} A & \stackrel{f}{\longrightarrow} & Y \\ i \downarrow & & \downarrow {\phi_2} \\ X & ...
14
votes
1answer
488 views

Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?

In this question, The Chaz asks whether $G\times G\cong H\times H$ implies that $G\cong H$, where $G$ and $H$ are finite abelian groups. The answer is to his question is yes, by the structure theorem ...
0
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2answers
105 views

Presheaf on a topological space

Consider the category of all open sets of a given topological space where the morphism are inclusions,why one can see a Presheaf as a contravariant functor?
4
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0answers
97 views

For an arbitrary ring $R$ and a positive integer $n >1$, are the category of $R$-modules and the category of $M_n(R)$-modules isomorphic?

For an arbitrary ring $R$ and a positive integer $n >1$, are the category of $R$-modules and the category of $M_n(R)$-modules isomorphic? Here, $M_n(R)$ denotes the $n$ plus $n$ matrices over the ...
8
votes
2answers
428 views

Can we say that $V \cong V^*$ is not natural?

Can we say that an isomorphism $V \cong V^*$ is not natural? It seems so intuitively, but formally the notion of a natural transformation between a functor and a cofunctor is not defined (or is it?).
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votes
1answer
101 views

Labeled and unlabeled categories

When one talks about the category $V_K$ of vector spaces over a field $K$ and considers the dual functor $D$ which maps a vector space $V$ to its dual $V^{*}$ I believe to have in mind something like ...
6
votes
3answers
671 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
0answers
241 views

Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
7
votes
1answer
287 views

Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon ...
26
votes
5answers
2k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
6
votes
1answer
169 views

restriction map in a Sheaf of $\mathcal{O}_X$ modules

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. I have trouble understanding the restriction map in the definition of the sheaf of $\mathcal{O}_X$ modules. Explicitly, let ...
12
votes
1answer
498 views

In an additive category, why is finite products the same as finite coproducts?

In an additive category, why is finite products the same as finite coproducts? This is relatively easy to prove when the category is R-mod, but my intuition/creativity fails to see how the method can ...
20
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1answer
452 views

cones in the derived category

If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' ...
3
votes
1answer
264 views

Class models in set theory and category theory

Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, ...
5
votes
1answer
374 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 ...
20
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2answers
933 views

How can there be alternatives for the foundations of mathematics?

How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of ...
8
votes
3answers
504 views

The identity morphism in $\mathbf{Set}$ is the identity function

I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. ...
6
votes
3answers
719 views

Is learning haskell a bad thing for a beginner mathematician?

Haskell is a programming language which uses some concepts from category theory like functor, monad, etc. My question is: Learning intuitive concepts about category from Haskell will ruin my intuition ...
39
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8answers
5k views

When to learn category theory?

I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
5
votes
0answers
200 views

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of ...
15
votes
1answer
642 views

Adjoint functors as “conceptual inverses”

The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other. For example, the forgetful functor "ought ...
6
votes
3answers
2k views

Special arrows for notation of morphisms

I've stumbled upon the definition of exact sequence, particularly on Wikipedia, and noted the use of $\hookrightarrow$ to denote a monomorphism and $\twoheadrightarrow$ for epimorphisms. I was ...
13
votes
2answers
941 views

Why is every abelian group the direct limit of its finitely generated subgroups?

I'm taking classes in homological algebra now, and the book (together with the lecturer) seem to assume more category theory than I already know. A "fact" that is used freely in the book ...
1
vote
1answer
94 views

embedding a k-linear category in an additive category

I am reading an article and it is written there: If A is a k-linear category (possibly without direct sums) we can embed it in the additive category A × N, where a morphism (x,m) → (y, n) is an n × m ...
2
votes
1answer
403 views

Inverse limit in the category of sets

This is in reference to page 153 of the notes found here http://www.math.lsa.umich.edu/~hochster/614F10/614.pdf In the second paragraph, the author is considering inverse limit of a family of subsets ...
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0answers
161 views

prove isomorphism in distributive category

I have a question, I want to prove cartesian closed category with finite coproduct is distributive category. but do not know how to prove A*(B+C)~A*B+A*C is isomorphism from the property of ...
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0answers
232 views

What is the name for a “lifted” representable functor?

Suppose $F : C \to D$ is a functor between concrete categories. If $U_D : D \to \text{Set}$ and $U_C : C \to \text{Set}$ are the corresponding forgetful functors to $\text{Set}$, then it's natural to ...
15
votes
3answers
1k views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
2
votes
2answers
213 views

$(-1)\otimes (-1) \cong I$

Is there a monoidal category $\mathcal C$ whose unit object is $I$ (i.e. $I\otimes A\cong A\cong A\otimes I$ for all $A\in \text{Ob}_\mathcal C$), with an object "$-1$" such that $$ ...
10
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2answers
515 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 ...
14
votes
1answer
548 views

Categorification of $\pi$?

Is there a categorification of $\pi$? I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my ...
5
votes
1answer
389 views

Pullback-stability for epimorphisms

In category theory, you see the idea of a class of epimorphisms being stable under pullback. For example, in a regular category, the class of regular epimorphisms is closed under pullback. Every ...
2
votes
1answer
399 views

A question in “Conceptual Mathematics”

In Conceptual Mathematics 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof. "... This also follows from the appropriate universal mapping properties, ...
2
votes
1answer
189 views

Defining “subfunctors”

I have a functor $F\colon \mathbf{Rng}\to\mathbf{Grp}$, and a correspondence on objects which assigns to every group $F(R)$ a suitable subgroup $G_o(R)\subseteq F(R)$. Is there a way to turn $G$ into ...
5
votes
0answers
96 views

How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”

Thanks for your attention, I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras, I cannot prove this ...
8
votes
1answer
352 views

Is $\mathbf{Grp}$ a concrete category?

Is $\mathbf{Grp}$ a concrete category? I thought it is, but then the group of symmetries of a square and the quaternion group are both of the order 8, and they are not isomorphic as groups. But sets ...
3
votes
4answers
316 views

Chromatic Filtration of Burnside Ring

I just attended a seminar on the chromatic filtration of the Burnside ring. I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration ...
4
votes
0answers
99 views

Practical approaches to working with nonplanar commutative diagrams?

The 4-associahedron is the 4-dimensional version of Mac Lane's pentagon diagram. If you look at Trimble's notes on tetracategories, you can see the obvious difficulty in working with such a diagram ...
4
votes
2answers
255 views

Category of sets

Let C be a category of sets, which has objects all sets and arrows all functions, with usual identity functions and the usual composition of functions. For any set S, the assignment s-s for all s in S ...
2
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0answers
104 views

Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
9
votes
2answers
679 views

Why is Top a model category?

Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy ...
6
votes
2answers
745 views

What is the difference between Categories and Relations?

For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer. Category: A ...
5
votes
2answers
230 views

Where is the well-pointedness assumption of ETCS used in everyday math?

Where is the well-pointedeness assumption of the Elementary theory of the category of sets (Lawvere's category-theoretic axiomatization of set theory) used in everyday math? Specifically, if you have ...
3
votes
1answer
233 views

Is restriction of scalars a pullback?

I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback". I don't understand why this might be the case. In particular, consider a ...
4
votes
1answer
230 views

Yoga of localization in categories?

In the derived category $D(C)$ of an abelian category $C$, one formally inverts quasi-isomorphisms. In the context of model categories, one inverts weak equivalences. What does one gain by doing ...
15
votes
4answers
2k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
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2answers
425 views

Morphisms in the category of natural transformations?

I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject. The resources I've seen all take the following approach: 0) A category is a ...
10
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3answers
2k 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.
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votes
1answer
351 views

Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
2
votes
1answer
128 views

Is the empty subcategory thick, localizing, topologizing, etc

Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the ...