There is something about category theory that just puts me to sleep. I cannot count the number of times I picked up a category theory text, full of best intentions, started reading, and ... woke up a few hours later.
Dozens of diagrams pointing here and there don't help. It is somehow as if the abstractness of category theory is abstract in the wrong way for me.
It’s not just you. Category theory has been called “abstract nonsense” for a very long time and even referred to as such by Saunders Mac Lane [1], cofounder of the discipline itself! The subject is just very difficult to motivate because it’s so abstract that it’s hard to see the relevance of its results.
Yet its origin seems to be something quite concrete and practical.
For example, just recently I came across this text:
"Foundations of Algebraic Topology", by Eilenberg and Steenrod. Its preamble is highly readable and engaging, see below. We have a topology and compute some algebraic structure from it. Sounds easy!
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The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest and most extensively developed portion of algebraic topology, and may be regarded as the main body of the subject. The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized.
It is assumed that the reader is familiar with the basic concepts of algebra and of point set topology. No attempt is made to axiomatize these subjects. This has been done extensively in the literature. Our achievement is different in kind. Homology theory is a transition (or function) from topology to algebra. It is this transition which is axiomatized.
Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems.
In this respect, homology theory parallels analytic geometry. How ever, unlike analytic geometry, it is not reversible. The derived algebraic system represents only an aspect of the given topological system, and is usually much simpler. This has the advantage that the geometric problem is stripped of inessential features and replaced by a familiar type of problem which one can hope to solve. It has the disadvantage that some essential feature may be lost. In spite of this, the subject has proved its value by a great variety of successful applications.
Our axioms are statements of the fundamental properties of this assignment of an algebraic system to a topological system. The axioms are categorical in the sense that two such assignments give isomorphic algebraic systems.
Eilenberg and Steenrod is great, but if you're not quite ready for algebraic topology, Mac Lane and Birkhoff's _Algebra_ [1] is an introductory abstract algebra text that makes well-motivated use of category theory throughout. From the preface:
…we hold that the general and abstract ideas needed should grow naturally from concrete instances. With this in view, it is fortunate that we do not need to begin with the general notion of a category. The most basic category is the category whose objects are all sets and whose morphisms are all functions (from one set to another); hence we can start Chapter I with sets — more accurately, with sets, functions, and the composition of functions — as the fundamental materials. On this background, Chapter II introduces the integers as the most basic example of an algebraic system. All the other categories which we need are quite "concrete" ones — each object A in the category is a set (with some structure), and each morphism from an object A to an object B in the category is a function (one which preserves the structure) on the set A to the set B. Hence we can give in Chapter I an easy, explicit definition of a "concrete" category, leaving the full treatment of the more general notion of a category to Chapter XV. In the same spirit, the idea fundamental to the notion of adjoint functor turns out to be the simple one of a "universal" construction. This idea, introduced in Chapter I for sets and in Chapter II for other concrete categories (such as monoids and lattices), is then developed with successive examples throughout the subsequent chapters.
What kind of mathematical background do you have? Despite category theory itself being entirely independent of anything else (it has pretty much no formal prerequisites for its study), without an undergraduate level knowledge of pure mathematics I imagine it’s quite hard to appreciate what’s going on and what the value of any of it is. And the deeper you get, the more it attempts to model and simplify even more ‘advanced’ branches of the subject.
Category theory doesn't have to be presented in such a diagram-heavy way. It's just that the people who love category theory also love diagrams.
I don't know one that avoids diagrams as much as possible, but the book "Abstract and Concrete Categories" is not too diagram heavy. I think the PDF is legally available online.
My main problem with these books (this one also) is that they don't take me to anything interesting I can do with categories quickly enough before I fall asleep.
It seems I have to wade through hundreds of trivial examples first.
Maybe I should write my own book about categories, that might keep me awake long enough.
Oh, maybe that is why there are so many books about category theory!
A lot of category theory is just identifying that a bunch of similar elementary results across algebra and topology can be stated and proved once in a common framework. The examples in ACC serve to show the different guises the elementary results take in different settings, as well as a source of counterexamples. I would guess this is not to your taste.
The only nontrivial theorems of general interest are the Adjoint Functor Theorems, which give very general existence proofs for universal constructions. For that, the chapter in Saunders Maclane's book is probably better.
Though it could be that none of this is worth your time. Category theory inspires intense enthusiasm from some people, but outside of a few areas of math it's not strictly necessary.
Comments
There is something about category theory that just puts me to sleep. I cannot count the number of times I picked up a category theory text, full of best intentions, started reading, and ... woke up a few hours later.
Dozens of diagrams pointing here and there don't help. It is somehow as if the abstractness of category theory is abstract in the wrong way for me.
It’s not just you. Category theory has been called “abstract nonsense” for a very long time and even referred to as such by Saunders Mac Lane [1], cofounder of the discipline itself! The subject is just very difficult to motivate because it’s so abstract that it’s hard to see the relevance of its results.
[1] https://en.wikipedia.org/wiki/Abstract_nonsense
Yet its origin seems to be something quite concrete and practical.
For example, just recently I came across this text: "Foundations of Algebraic Topology", by Eilenberg and Steenrod. Its preamble is highly readable and engaging, see below. We have a topology and compute some algebraic structure from it. Sounds easy!
------------------------------
The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest and most extensively developed portion of algebraic topology, and may be regarded as the main body of the subject. The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized. It is assumed that the reader is familiar with the basic concepts of algebra and of point set topology. No attempt is made to axiomatize these subjects. This has been done extensively in the literature. Our achievement is different in kind. Homology theory is a transition (or function) from topology to algebra. It is this transition which is axiomatized. Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems. In this respect, homology theory parallels analytic geometry. How ever, unlike analytic geometry, it is not reversible. The derived algebraic system represents only an aspect of the given topological system, and is usually much simpler. This has the advantage that the geometric problem is stripped of inessential features and replaced by a familiar type of problem which one can hope to solve. It has the disadvantage that some essential feature may be lost. In spite of this, the subject has proved its value by a great variety of successful applications. Our axioms are statements of the fundamental properties of this assignment of an algebraic system to a topological system. The axioms are categorical in the sense that two such assignments give isomorphic algebraic systems.
Eilenberg and Steenrod is great, but if you're not quite ready for algebraic topology, Mac Lane and Birkhoff's _Algebra_ [1] is an introductory abstract algebra text that makes well-motivated use of category theory throughout. From the preface:
…we hold that the general and abstract ideas needed should grow naturally from concrete instances. With this in view, it is fortunate that we do not need to begin with the general notion of a category. The most basic category is the category whose objects are all sets and whose morphisms are all functions (from one set to another); hence we can start Chapter I with sets — more accurately, with sets, functions, and the composition of functions — as the fundamental materials. On this background, Chapter II introduces the integers as the most basic example of an algebraic system. All the other categories which we need are quite "concrete" ones — each object A in the category is a set (with some structure), and each morphism from an object A to an object B in the category is a function (one which preserves the structure) on the set A to the set B. Hence we can give in Chapter I an easy, explicit definition of a "concrete" category, leaving the full treatment of the more general notion of a category to Chapter XV. In the same spirit, the idea fundamental to the notion of adjoint functor turns out to be the simple one of a "universal" construction. This idea, introduced in Chapter I for sets and in Chapter II for other concrete categories (such as monoids and lattices), is then developed with successive examples throughout the subsequent chapters.
[1] https://archive.org/details/algebra00macl
What kind of mathematical background do you have? Despite category theory itself being entirely independent of anything else (it has pretty much no formal prerequisites for its study), without an undergraduate level knowledge of pure mathematics I imagine it’s quite hard to appreciate what’s going on and what the value of any of it is. And the deeper you get, the more it attempts to model and simplify even more ‘advanced’ branches of the subject.
Category theory doesn't have to be presented in such a diagram-heavy way. It's just that the people who love category theory also love diagrams.
I don't know one that avoids diagrams as much as possible, but the book "Abstract and Concrete Categories" is not too diagram heavy. I think the PDF is legally available online.
My main problem with these books (this one also) is that they don't take me to anything interesting I can do with categories quickly enough before I fall asleep. It seems I have to wade through hundreds of trivial examples first.
Maybe I should write my own book about categories, that might keep me awake long enough.
Oh, maybe that is why there are so many books about category theory!
Perhaps this lecture series by Bartosz Milewski is more your thing
https://youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa...
His enthusiasm and energy certainly helps to keep it entertaining
I have toyed with writing my own intro.
A lot of category theory is just identifying that a bunch of similar elementary results across algebra and topology can be stated and proved once in a common framework. The examples in ACC serve to show the different guises the elementary results take in different settings, as well as a source of counterexamples. I would guess this is not to your taste.
The only nontrivial theorems of general interest are the Adjoint Functor Theorems, which give very general existence proofs for universal constructions. For that, the chapter in Saunders Maclane's book is probably better.
Though it could be that none of this is worth your time. Category theory inspires intense enthusiasm from some people, but outside of a few areas of math it's not strictly necessary.
there are a few books that make it accessible to non mathematicians. Both by Eugenia Cheng, "How to Bake Pi" and "The Joy of Abstraction"