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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17answers
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39
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8answers
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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 ...
85
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5answers
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
14
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2answers
2k views

Yoneda-Lemma as generalization of Cayley`s theorem?

I came across the statement, that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations. How exactly does generalizes ...
5
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2answers
750 views

Adjoint functors

I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor ...
12
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2answers
717 views

Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a ...
35
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1answer
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Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
10
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1answer
545 views

Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
8
votes
2answers
301 views

Tensor product of monoids and arbitrary algebraic structures

Let $C$ be the category of algebraic structures of a certain type and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$ we have a functor $\mathrm{BiHom}(M,N;-) : C ...
5
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1answer
242 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
4
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1answer
220 views

Strictly associative coproducts

Background. This question belongs to evil mathematics. It is motivated by this question which links to a paper in which it is claimed that it is an open problem whether there exists strictly ...
1
vote
2answers
135 views

category of linear maps

Let $V,W$ be vector spaces. Let's define a category whose objects are linear map $f:V\to W$ and morphisms from $f$ to $g$ are pair of linear maps $(\alpha,\beta)$ where $\alpha:V\to V,\beta :W\to W$ ...
3
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1answer
91 views

Do pullbacks commute with filtered limits in this sense?

Let $A_n, B_n, C_n$ be directed systems in some abelian category. Denote by $A \times_C B$ the fibre product of $A$ and $B$ over $C$. Is it true that $(\varprojlim A_n) \times_{\varprojlim C_n} ...
26
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5answers
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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 ...
27
votes
4answers
2k views

Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
19
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3answers
1k 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 ...
21
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1answer
221 views

Rigidity of the category of fields

Let's call a category rigid if every self-equivalence is isomorphic to the identity. For example, $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{CRing}$ (MO/106838), $\mathsf{Top}$ ...
33
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1answer
796 views

Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$

A long time back, I wondered what functions other than integer polynomials on $\mathbb{N}$ (or $\mathbb{Z}$) satisfied the property: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
17
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1answer
2k views

Quotient objects, their universal property and the isomorphism theorems

This is a question that has been bothering me for quite a while. Let me put between quotation marks the terms that are used informally. "Quotient objects" are always the same. Take groups, abelian ...
10
votes
3answers
535 views

What are the epimorphisms in the category of Hausdorff spaces?

It appears to be the case that the epimorphisms in $\text{Haus}$ are precisely the maps with dense image. This is claimed in various places, but a comment on my blog has made me doubt the source I got ...
15
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1answer
651 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 ...
14
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3answers
2k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...
13
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2answers
711 views

what are the product and coproduct in the category of topological groups

I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.
8
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1answer
852 views

Unique up to unique isomorphism

If an object $X$ has a non-trivial automorphism $g$, for any isomorphism $f$ with an object $Y$ there is another isomorphism $f \circ g$ between $X$ and $Y$, so there is not a unique isomorphism ...
6
votes
2answers
315 views

Is gcd the right adjoint of something?

In his answer link to the question whether $a|m$ and $a+1|m$ implies $a(a+1)|m$, Bill Dubuque takes a detour to derive the equality $$ \gcd(a,b)=ab/\mathrm{lcm}(a,b) $$ from the universal property of ...
6
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2answers
785 views

Learning to think categorically (localization of rings and modules)

I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal ...
10
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1answer
543 views

The categories Set and Ens

I've recently started reading Categories for the Working Mathematician and I'm a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like ...
7
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3answers
613 views

“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?

Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if ...
10
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1answer
467 views

Monic (epi) natural transformations

Let $C$ and $D$ be categories and let $F : C \rightarrow D$, $G : C \rightarrow D$ be two functors such that they are either both covariant or both contravariant. Under what most general hypotheses is ...
7
votes
4answers
228 views

Categorial definition of subsets

I cannot see why this definition http://en.wikipedia.org/wiki/Subobject is equivalent to subsets in the category of sets. I am confused by the following facts, 1) the partial order is defined ...
4
votes
1answer
318 views

Equivalent definition of exactness of functor?

I'll use the following definition: (Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences. Now I'd like to prove the following (not entirely sure it's ...
2
votes
2answers
435 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...
11
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1answer
341 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
5
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1answer
353 views

Excessive use of the Yoneda lemma

In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof: If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma. I understand ...
5
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3answers
270 views

Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...
2
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1answer
122 views

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := ...
2
votes
2answers
625 views

Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category

I've been trying to find a proof that the pullback functors in a locally cartesian closed category have right adjoints (used to model the notion of indexed product inside a category (rather than ...
1
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1answer
372 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
5
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1answer
206 views

Injective Cogenerators in the Category of Modules over a Noetherian Ring

Let $R$ be a Noetherian ring and let $\mathcal{A}$ be an injective $R$-module. The injectivity of $\mathcal{A}$ is equivalent to the exactness of the functor $Hom_R(-,\mathcal{A})$, i.e. whenever we ...
3
votes
2answers
697 views

What is the categorical diagram for the tensor product?

The title says it all: what is the diagram that defines a tensor product? (I'm using the term diagram here in the technical sense it has in category theory.) Edit: This question was motivated by the ...
22
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4answers
627 views

A bestiary about adjunctions

What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying ...
40
votes
6answers
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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, ...
31
votes
7answers
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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 ...
31
votes
5answers
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What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
18
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5answers
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What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
17
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1answer
2k views

Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, ...
15
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3answers
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What is the origin of the expression “Yoneda Lemma”?

Thank you very much in advance for telling where the expression “Yoneda Lemma” comes from. EDIT 1. On page -14 of Reprints in Theory and Applications of Categories, No. 3, 2003. Abelian Categories, ...
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3answers
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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 ...
14
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5answers
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Concrete examples of 2-categories

I've been reading some of John Baez's work on 2-categories (eg here) and have been trying to visualize some of the constructions he gives. I'm interested in coming up with 'concrete' examples of ...
14
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3answers
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Comparing Category Theory and Model Theory (with examples from Group Theory).

The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple: (Q1)If someone write "the category ...