Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seeming different areas of mathematics.

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Categorical introduction to Algebra and Topology

At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book Algebra by Serge Lang. I have read the chapters on groups and rings, but then my ...
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What's the deal with empty models in first-order logic?

Asaf's answer here reminded me of something that should have been bothering me ever since I learned about it, but which I had more or less forgotten about. In first-order logic, there is a convention ...
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Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
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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 ...
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Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
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When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then $D(...
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Is Set “prime” with respect to the cartesian product?

(Motivated by Stefan Perko's question here) Suppose $C, D$ are two categories such that $\text{Set} \cong C \times D$. Is either $C$ or $D$ necessarily equivalent to the terminal category $1$? I ...
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Mnemonic for the fact that a right(left) adjoint functor preserves limits(colimits)

A right adjoint functor preserves limits. Dually a left adjoint functor preserves colimits. I often forget which is which. Of course, you can look up a book on category theory or use internet. But it'...
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How is Category Theory used to study differential equations?

I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions \begin{equation} s,t:\mathbb{R}^3\...
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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 ...
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What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
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Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V \...
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what is the difference between functor and function?

As it is, what is the difference between functor and function? As far as I know, they look really similar. And is functor used in set theory? I know that function is used in set theory. Thanks.
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Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
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What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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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 2-...
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Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
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A Category Theoretical view of the Kalman Filter

Some basic background The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical ...
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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 $B \to A$, ...
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For which categories we can solve $\text{Aut}(X) \cong G$ for every group $G$?

It is usually said that groups can (or should) be thought of as "symmetries of things". The reason is that the "things" which we study in mathematics usually form a category and for every object $X$ ...
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Why is every category not isomorphic to its opposite?

This is a beginner category theory question: I'm trying to wrap my head around the fact that we do not have $\mathbf{Sets} \cong \mathbf{Sets}^\mathsf{op}$, i.e., the category of sets is not ...
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Examples of categories where morphisms are not functions

Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
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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 ...
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Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over $\mathbb{R}$...
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Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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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 ...
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Additive category that is not abelian

What is a simple example, without getting into the mess of triangulated categories, of an additive category that is not abelian?
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Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers

Let $\hat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$. Since $\hat{\mathbb{Z}}$ is the inverse limit of the rings $\mathbb{Z}/n\mathbb{Z}$, it's a subgroup of $\prod_n \mathbb{Z}/n\mathbb{...
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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, ...
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Maps in Opposite Categories

Given some category ${\mathcal C}$, then the opposite category will consist of the same objects with the morphisms "turned around." Given $f:A\rightarrow B$, for $A,B$ objects of ${\mathcal C}$, then ...
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Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what I'm ...
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Which insights are particularly simple to get through category theory?

I'm trying to motivate myself to learn category theory, but I don't have a clear view of what would be the benefit. I've heard that it can be used to find similitudes between mathematical theories, ...
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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 ...
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Has category theory solved major math problems?

I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ?, i.e has ...
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Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
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How different can equivalent categories be?

Some pretty big results in mathematics can be expressed as equivalence of categories (or rather, as adjunctions). The usual definition $FG\cong 1,GF\cong 1$ is easily justifiable in category theory ...
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Group Theory via Category Theory

I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for ...
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Can “Taking algebraic closure” be made into a functor?

I am now confused with such problem as title goes. To be exact, the problem is Does there exist a functor from $A:\mathsf{Field}\to \mathsf{Field}$ with a natural transformation from identity ...
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Can it happen that the image of a functor is not a category?

On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
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Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
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What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
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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 ...
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Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
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If adjunction arises everywhere, where is it in the fundamental theorems?

MacLane's slogan "adjunction arises everywhere" is widely known, and adjunction has been identified as a key concept (maybe the key concept?) in category theory, eg, in the books by Goldblatt Topoi, ...
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Calculus and Category theory

Quick question: Is it possible to differentiate a function with respect to another function, or is it limited to a particular variable? I tried thinking around how to make this question make sense, ...
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Category Theory vs. Universal Algebra - Any References?

After seeing the answer to the question Category theory, a branch of abstract algebra, I would like to ask Are there literature discussing the difference/indifference/comparison between category ...
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In category theory why is a right adjoint not a left adjoint?

I'm learning basic category theory and teaching myself about adjoints. The definition I have is that an adjunction between $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ is a ...
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Yoneda Lemma Exercises

Can you please suggest some (relatively simple) exercises to practice the use of the Yoneda Lemma? Harder exercises are welcome too, but I would like to start with simpler ones. The answers to this ...
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Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
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On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...