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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75
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22answers
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15
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1answer
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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 ...
113
votes
6answers
7k views

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 ...
52
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8answers
7k 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 ...
17
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1answer
478 views

Analogue of the Cantor-Bernstein-Schroeder theorem for general algebraic structures

The Cantor-Bernstein-Schroeder theorem states that if there are two sets $A$ and $B$ such that there exist injective (alternatively, surjective, assuming choice I think) maps $A \to B$ and $B \to A$, ...
18
votes
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 ...
18
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2answers
983 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 ...
6
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2answers
929 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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3answers
754 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 ...
6
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2answers
154 views

''Labelling discrimination'' for objects in a category

I am not particularly used to the category theory thinking paradigm and there are certain statements that I am used to making in the set theoretic modelling approach that I would like to characterize ...
8
votes
4answers
282 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 ...
7
votes
2answers
168 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
31
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5answers
3k 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 ...
29
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4answers
3k 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 ...
37
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1answer
696 views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
27
votes
6answers
4k views

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 ...
15
votes
3answers
644 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' ...
41
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1answer
3k views

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 ...
8
votes
1answer
1k 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 ...
4
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1answer
137 views

Is it necessary for the homsets to be disjoint in a category?

In defining a (locally small) category, some books include the condition that if $A\neq C$ or $B\neq D$, then $\hom(A,B)$ and $\hom(C,D)$ are disjoint sets. Is it necessary?
8
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2answers
369 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
331 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 ...
3
votes
2answers
90 views

Categorical Pasting Lemma

If I'm not mistaken, the pasting lemma for a two-element open cover $X=A\cup B$ of a topological space is equivalent to saying that the following square is a pushout in $\mathsf{Top}$: ...
4
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1answer
258 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
150 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
votes
1answer
194 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} ...
32
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4answers
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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 ...
26
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4answers
913 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 ...
22
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1answer
3k 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 ...
37
votes
7answers
1k 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 ...
24
votes
1answer
360 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), $\mathsf{Top}$ ...
39
votes
1answer
932 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: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
20
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1answer
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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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2answers
939 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.
14
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3answers
3k 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 ...
15
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1answer
739 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 ...
16
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4answers
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Reference request: compact objects in R-Mod are precisely the finitely-presented modules?

Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the ...
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4answers
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Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an ...
9
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4answers
865 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 ...
7
votes
1answer
667 views

Infinite coproduct of rings

I just learned from Wikipedia that coproduct of two (commutative) rings is given by tensor product over integers, and that coproduct of a family of rings is given by a "construction analogous to the ...
7
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2answers
339 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 ...
7
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3answers
952 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 ...
11
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1answer
649 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 ...
12
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1answer
637 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 ...
12
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1answer
519 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 ...
8
votes
3answers
2k views

Epimorphism and Monomorphism = Isomorphism?

It seems to be that if a map is both an epimorphism and a monomorphism, it is not necessarily the case that it is an isomorphism. However, in the category of sets, if a map is both an epimorphism and ...
4
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1answer
449 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 ...
3
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2answers
636 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 ...
3
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2answers
342 views

Verify this is a category

I'm working on a question in an intro algebraic topology book: Verify that the "arrow category" is a category. That is, if $\mathcal{C}$ is a category, then show that the following construction gives ...
3
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2answers
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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 ...