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 preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

A category $\mathcal{C}$ consists of a collection $\mathrm{Ob}(\mathcal{C})$ of objects and for any two objects $X$ and $Y$ a collection of arrows (often called morphisms) denoted $\mathrm{Hom}(X,Y)$ such that

  1. Every object $X$ comes equipped with an identity morphism $\mathbf{1}_X \in \mathrm{Hom}(X,X)$.
  2. For morphisms $f \in \mathrm{Hom}(X,Y)$ and $g \in \mathrm{Hom}(Y,Z)$, there exists a composite morphism $g\circ f \in \mathrm{Hom}(X,Z)$.

Many classes of mathematical objects can be collected and described as a category. For example:

  • $\text{Set}$ is the category with sets as objects and functions as morphisms. A subcategory of $\text{Set}$ is $\text{FinSet}$, the category of finite sets.
  • $\text{Grp}$ is the category of groups with morphisms given by group homomorphisms because these are the functions on groups that preserve their structure. $\text{Grp}$ also has a ubiquitous subcategory $\text{Ab}$, the category of abelian groups.

And the relationship between these mathematical objects, either within their own category or with another category, can be described in a coherent way using category theory. In particular many constructions within a category can be described as a limit or colimit, and many relationships between different categories can be described as a functor.

For more reading on category theory, see

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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 the guys in my office, and despite a very shady…
Asaf Karagila
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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 Approach to Integration". The Abstract: "We…
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Good books and lecture notes about category theory.

What are the best books and lecture notes on category theory?
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Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use category theory. Consider the category CpltMet in…
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What is category theory useful for?

Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it is to formalize. As far as I can tell, it sort of…
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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 about groups and modular arithmetic. Is it too…
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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, I started taking some functional analysis courses…
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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 double duals; they are isomorphic to their duals as…
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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 tedious details concerning duality. Suppose that $U:…
Hans-Peter Stricker
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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 inverse image of sheaves, spec and global…
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Is it possible to formulate category theory without set theory?

I have never understood why set theory has so many detractors, or what is gained by avoiding its use. It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can…
Matt Calhoun
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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 everything until we covered the Yoneda Lemma,…
Chris Taylor
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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal properties. It was an interest little shared by…
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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 Mathematics. For the sake of clarity I would like you to…
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What use is the Yoneda lemma?

Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse concepts. Many of the tools of category theory…
Alex Becker
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