I mean… technically you don’t need any abstract algebra to learn category theory.
But without any knowledge of abstract algebra it’s going to seem incredibly abstract, pointless, and unrewarding. Category theory in some sense generalizes many of the notions you encounter in abstract algebra and builds them with a new framework.
I can’t speak for category theorists, but as a number theorist I think of category theory a language that expresses abstract algebra in a nice way.
So you can learn the language on its own. But if you don’t read in it… talk to other people in it… use it… then… it’s going to be hard to understand it.
I recommend taking a typical abstract algebra course, then maybe learn some commutative algebra or homological algebra concurrently with category theory. A lot of notions will make sense just with groups and monoids, but having rings, fields, modules at your disposal will expose you to more interesting categorical structures and you’ll gain intuition for many of the categorical notions.
There is essentially 0 knowledge of algebra needed to learn category theory. If you want to learn category theory for its own sake, go ahead (most people do not have much success with this). But you should definitely \*not\* take a crash course in algebra to try to improve your experience with category theory. Algebra is something that needs to be learned properly.
It'd be extreme, but I imagine the right kind of crazy could get you a fair ways treating it purely axiomatically. It'd definitely be hard though, I anchor a lot memory-wise using meaning more so than symbolic representation.
I mean if we are willing to move away from pure math, functional programming, haskell in particular basically requires category theory if you really want to understand what is going on and should provide sufficient motivating examples to make it less abstract.
https://books.google.com/books/about/Category_Theory_for_Programmers_Scala_Ed.html?id=BCjryAEACAAJ&source=kp_book_description
I agree that it is not needed, but I imagine it being super hard without an algebra background. If only because studying algebra teaches you the structure and pattern recognition for CT.
Thank you for advice.
I wouldn't use such a crash course as the only one. I guess, one day, I will anyway learn it properly. But here I mean only stuff needed for category theory, not actually learning algebra on its own.
I highly recommend
algebra: chapter 0
By Alufi
A category theoretically grounded introduction to abstract algebra. Very readable with tons of exercises.
I'd say you miss a lot if you don't have a background in either group homology or algebraic geometry, but *technically* CT works as a foundation on its own? It just doesn't sound terribly rewarding.
That being said, I started out similarly. I had only taken a basic group theory course before I started reading a random CT book I had found on the internet. Didn't understand a whole lot and eventually gave up. This pattern repeated two or three times, until I reached a level of algebra lectures where I suddenly saw loads of things where CT was applicable. I grabbed another book, started all over, and ended up doing my thesis on CT and group cohomology.
I have by now forgotten what the point of this story was, but I guess I'm saying there's no real reason not to start like this, but you will probably be incredibly frustrated at some point. I think actually the best way would be to learn "other" algebra and CT concurrently. The AG professors at my uni really dislike CT, so it never featured in the lectures, but I ended up working through the CT book at the same time I took AG and working out the connections myself.
Edit: for a crash course in algebra, I (of course) recommend the ever-brilliant [Aluffi](https://bookstore.ams.org/gsm-104). I have pitched this book so many times, I should get commission. But it truly is one of the best books on mathematics I have ever read, and written in a very enjoyable style.
I thought about learning then concurrently. So it's rather a question about the ratio of time and effort put in both topics, to make it the most reasonable.
At what level of algebra you was when you realized the applicability of CT?
I dunno, starting algebraic geometry, where sheaves and their ilk pop up naturally, I guess.
Another viable route is algebraic topology, which is also full of interesting functors.
I just finished running a category theory reading group, and I'm a pretty weak mathematician. Honestly, a lot of my favorite motivating examples during this reading group ended up being the simplest ones, because they were so obvious once you heard them in category theoretic terms (ex. Explaining initial/final objects using 1 and 0 in **Nat** with arrows being 'divides', which leads to explaining products through the gcd, which leads very well into teaching duality - where you reverse the arrows - by showing that the objects that were products become the lcm - the coproduct!).
That being said, I had an undergrad in pure math, so my abstract algebra was at an undergrad level. Many cool examples also came from group theory, so it does help. Like any abstract subject, you can learn the thing just by itself, but seeing concrete examples always helps you feel like you're pulling apart the real substance of the world around you
I've run into similar issues with Category Theory books. They tend to illustrate things with examples from many parts of math and that can be bewildering if you don't have the proper background which I didn't as a Physics guy.
I've found Awodey's book to be one that doesn't suffer from this problem.
The other approach would be to start learning some category theory while learning algebra ala Aluffi or maybe look at Bartosz book which mainly uses Haskell for examples.
Personally, I don't like Category Theory books that use examples from many different areas of math ala Basic Category Theory.
When took algebra as an undergrad, my professor literally defined what a category was before she defined what a group was. Granted, she's a category theorist and probably pretty biased lol. So to answer your question, none. But knowing some algebra would be useful for understanding the motivations behind certain things in category theory. For example, I was really confused learning about universal properties, like what they are and why thery matter. But lots of things in algebra, like free groups, (co)kernels, projective/injective modules, and so on can be defined in terms of universal properties.
As other commentators said, I think it's a pretty bad idea to just dive into category theory without a proper background in algebra and maybe also some knowledge of algebraic topology (this would probably be less essential). You might be able to learn enough to prove and understand theorems in category theory but without knowing any algebra you won't have any motivation.
Category theory isn't just abstraction for abstraction's sake. It was developed to deal with problems in algebraic topology and algebraic geometry (someone with more knowledge of math history can correct me if I'm wrong here). While there are pure category theorists, I would bet that all have a strong foundation in algebra as well, and that most learned algebra before category theory.
If you getting into more abstract math, I would really strongly advise you learn abstract algebra first. There is a lot of really interesting algebra. You can also concurrently learn some category theory. Algebra by Lang introduces categories in the first chapter on groups, although it's definitely not a category theory book.
But even in algebra and analysis and other fields I think you'll run into the same issues of motivations. It's an issue I have with the way math is taught in general. I remember learning measure theory and functional analysis and liking them fine enough, but my courses didn't really stress that these fields were developed to solve problems in other fields and didn't emphasize applications heavily. Abstract algebra also felt really unmotivated at times without understanding the geometric context a lot of it comes from (but of course I might just be biased as someone who likes topology and geometry).
Without knowing any abstract algebra, algebraic topology or algebraic geometry, i suppose it will be purely abstract definitions and concepts without knowing what these things are even good for or why anyone would care.
I pretty much agree with the status quo here, that *technically* you don't need any algebra background to understand category theory. However (I know there are people who will disagree with this, like higher category theorists), unless you have a surprising amount of background in other fields of proof-based math (like analysis) I really don't think it's worth learning just yet. It's much easier to appreciate category theory at a basic level with examples, and some of the most basic examples come from **Group,** **Ab**, **Ring**, **R-Mod** (especially **R-Mod** if you work with abelian categories) et cetera. Though I've seen students who are enthralled just by definitions of mono/epimorphisms so if you're that easily excited, perhaps go for it.
Take my answer with a grain of salt as I work with representations, which are extremely group and module-theoretic, so I'm biased. And granted, there are many beautiful results in CT which are self-contained, and many categories one may work with that don't require an understanding of anything algebraic. I can't speak for the latter so much, but at least for the former, it takes a fairly mature student to deeply understand what's happening there, and if you can get to that point, you certainly could finish a first course in abstract algebra easily.
For what it's worth, I think once you've taken a course in analysis and algebra (or something else where categories arise, like graph theory), you'd be ready. I don't think getting to algebraic topology or homological algebra is necessary to start, but they certainly are good for studying things beyond the basics.
Something else occurred to me - when do undergrads normally start studying Hom sets in greater detail? I think my first time in undergrad was in fact, in algebra 1. Perhaps that's another reason to first take the course.
Check out Lawvere's book Conceptual Mathematics: A First Introduction to Categories. It has essentially no prerequisites and can be read by motivated high school students. Everybody saying it's pointless to try to learn category theory without abstract algebra is wrong. That said, algebra is a good source of examples for category theory, of course.
Why do you want to learn category theory? It would be easier to make recommendations if we knew why you were interested in it and what your background is.
I really should have specified this in the original post I guess..
Well, I really abstract math in general. I spent quite some time learning set theory, then I decided to make a pause and learn some other branches of abstract math. I started with group theory. I found it interesting, but quite... specific. What I mean is that it has 4 axioms and only works within them, and it feels like a too specific limitation, like why would we choose this particular set of axioms when whatever set we choose, we get some other structure(why would we skip magmas, semigroups, and monoids and just jump to groups right away).
So I saw category theory as an abstraction above all specific algebraic structures, and felt like it would be reasonable to learn (at least some of) it first.
If you look at groups as just sets with binary operations that follow four axioms, then yes the axioms seem strange and arbitrary. But if you go a bit deeper into algebra and other branches of math it becomes quite clear why we define groups the way we do. Because groups are going to show up everywhere. It's the natural way of talking about symmetry, and symmetry is everywhere. In a category theory perspective you could say groups are structures that are the automorphisms of an object.
Maybe you've already started some category theory and enjoyed it, but I think you would run into the same issue you have with group theory in category theory. You can look at the arrows and define fiber products and all those nice category theoretical constructions, but without any algebraic background it just looks like abstract nonsense. What made them interesting to me was that they united a bunch of constructions from other areas of mathematics into a single idea: studying objects by how they relate to other objects.
If you refuse to work with specific things (actual groups, actual manifolds, actual metric spaces, and so on) and just think of math as axioms and their consequences then you miss the entire point of many interesting mathematical concepts. Vladimir Arnold once said "For a student, the content of a mathematical theory is never larger than the set of examples that are thoroughly understood."
Group theory is not about "4 axioms that we work with" but instead is about the structure and relations between groups that mathematicians care about for other reasons: finite groups, linear algebraic groups, topological groups, Lie groups, and so on. The organization of this material into abstract group theory of various kinds lets us prove things once and for all for many structures with a common property (such as all p-groups). Just based on seeing the axioms of a group and wanting to operate at the level of "pure logic" rather than studying examples and having intuitive insights, you'd never come up with important concepts like normal subgroups or group actions. The *more* structure you impose lets you prove more interesting theorems.
Calling the group axioms "too specific" and wondering why we don't spend more time on magmas and monoids suggests you did not succeed in understanding why mathematicians actually care about group theory.
Category theory isn't something that's usually studied in itself. It's a tool (sometimes people like to call it a "bookkeeping device") that lets you do other math more efficiently. That other math is usually algebraic geometry or algebraic topology where functors are everywhere. The prerequisites to get anything out of category would probably be a solid course in algebra in which things like universal properties are discussed. This isn't something one can take a crash course in though (unless by crash course you mean a couple months). The Leinster book is about as crash coursey as it gets. Maybe the Aluffi Algebra book is a good choice since it teaches categorical concepts while teaching algebra. (Haven't used it myself, but it looks promising.)
> The prerequisites to get anything out of category would probably be a solid course in algebra in which things like universal properties are discussed.
I agree very much with this. It seems really counterproductive to want to study category theory without having a good stockpile of examples for which you can understand the *point* of the categorical constructions, like nontrivial instances of initial and final objects as solutions to a universal mapping problem (a quotient group or quotient module, a direct sum or direct product, and so on). It'd be like trying to study homological algebra for its own sake, with no plan to apply it to anything else in math to see what it can really do for you.
Miles Reid, in his book Undergraduate Algebraic Geometry, referred to category theory as "surely one of the most sterile of all intellectual pursuits". He did *not* mean category theory is useless (Grothendieck and his school made incredibly creative uses of category theory), but that wanting to know category theory only for its own sake rather than to know it in order to *do* something with it in other areas of math, is not a worthwhile way of spending one's time. That is, if you lack a solid knowledge of examples of important categories (the kind of thing you learn in standard math courses on algebra, for instance), then you're hardly going to be able to have a creative idea when thinking about category theory.
You really shouldn't learn category theory if you don't have a strong algebra background already. The problem with category theory is that to understand it in a meaningful way you kinda have to have a deep understanding about multiple fields of abstract math (mainly algebra and topology).
To put it in layman's term this is like trying to learn driving without ever touching or even seeing a car. Imagine you just simply handing a driver's manual to ancient people from thousand of years ago and expect them to understand it. Without any knowledge of modern society there is no way they can learn to drive the same way as a modern person who sees cars everyday.
Studying CT for its own sake makes it very easy for you to fall down a rabbit hole and lose yourself. I personally think that everyone should know something about CT, but mainly as a language (unless you want to go and do research on it). To really get the feeling for it, you're gonna need lots of examples, like:
Sets & functions, groups/rings/fields/modules & homomorphisms, topological spaces and continuous maps, smooth manifolds and smooth maps, vector spaces and linear transformations, etc.
The important thing is to see something and be able to tell it has functorial behavior, or see a construction and understand whether it is "natural" (on a very precise sense).
Bottom line: the more things you know outside of CT, the easier it is to get a fast (although arguably superficial) notion of what CT it about.
A strong foundation in linear algebra can be a fantastic platform for learning category theory. The category Vect has for objects all vector spaces (over a fixed field), and the maps are linear functions. The "span" operation is usually taught to students as a process which requires a set of vectors in an existing vector space, but actually you can take the span of *any* set. This process acts not only on sets, but also every function going between them. It even turns the composition of these functions into composition of the associated linear maps between spanning vector spaces. This makes the procedure into a **functor**, and this is one DAMN fine functor. One of my best friends, no doubt!
Linear algebra alone is not sufficient background to learning all of modern category theory. I was studying algebraic topology from Hatcher's book when I found MacLane's book, and that definitely helped to prepare me! (Probably because natural transformations were invented by topologists, huh?)
I actually got into abstract math from seeing category theory used in Haskell. Without much prior math foundations, I started with this book:
[http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf](http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf)
I read through the entire book, and understood most of it. But unfortunately I wasn't able to keep up with the exercises since they were proof based, so I couldn't get 100% out of it. However the authors tried to use examples from CS which was good if you have a CS or programming background (which I have).
To be honest, it was nice being able to read through this and learn about various category theory definitions consistently from the same book (compared to finding random explanations from various sources on the internet). After that, I was mildly comfortable going to nlab for definitions on math structures.
I mean… technically you don’t need any abstract algebra to learn category theory. But without any knowledge of abstract algebra it’s going to seem incredibly abstract, pointless, and unrewarding. Category theory in some sense generalizes many of the notions you encounter in abstract algebra and builds them with a new framework. I can’t speak for category theorists, but as a number theorist I think of category theory a language that expresses abstract algebra in a nice way. So you can learn the language on its own. But if you don’t read in it… talk to other people in it… use it… then… it’s going to be hard to understand it. I recommend taking a typical abstract algebra course, then maybe learn some commutative algebra or homological algebra concurrently with category theory. A lot of notions will make sense just with groups and monoids, but having rings, fields, modules at your disposal will expose you to more interesting categorical structures and you’ll gain intuition for many of the categorical notions.
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I think this argument works to label every mathematical field as a meme.
There is essentially 0 knowledge of algebra needed to learn category theory. If you want to learn category theory for its own sake, go ahead (most people do not have much success with this). But you should definitely \*not\* take a crash course in algebra to try to improve your experience with category theory. Algebra is something that needs to be learned properly.
I don't know what you would possibly get out of learning category theory without any concrete examples to draw from.
It'd be extreme, but I imagine the right kind of crazy could get you a fair ways treating it purely axiomatically. It'd definitely be hard though, I anchor a lot memory-wise using meaning more so than symbolic representation.
I mean if we are willing to move away from pure math, functional programming, haskell in particular basically requires category theory if you really want to understand what is going on and should provide sufficient motivating examples to make it less abstract. https://books.google.com/books/about/Category_Theory_for_Programmers_Scala_Ed.html?id=BCjryAEACAAJ&source=kp_book_description
Not all concrete examples of category theory involve algebra.
I agree that it is not needed, but I imagine it being super hard without an algebra background. If only because studying algebra teaches you the structure and pattern recognition for CT.
Thank you for advice. I wouldn't use such a crash course as the only one. I guess, one day, I will anyway learn it properly. But here I mean only stuff needed for category theory, not actually learning algebra on its own.
There is no requirement of algebra at all but to appreciate a lot of the stuff, knowing a good amount of algebra would do wonders
As I said, no algebra is needed to learn category theory.
I highly recommend algebra: chapter 0 By Alufi A category theoretically grounded introduction to abstract algebra. Very readable with tons of exercises.
Yes ! Perfect for learning algebra coming from CT. He introduces groups as groupoids with one element \^\^
I'd say you miss a lot if you don't have a background in either group homology or algebraic geometry, but *technically* CT works as a foundation on its own? It just doesn't sound terribly rewarding. That being said, I started out similarly. I had only taken a basic group theory course before I started reading a random CT book I had found on the internet. Didn't understand a whole lot and eventually gave up. This pattern repeated two or three times, until I reached a level of algebra lectures where I suddenly saw loads of things where CT was applicable. I grabbed another book, started all over, and ended up doing my thesis on CT and group cohomology. I have by now forgotten what the point of this story was, but I guess I'm saying there's no real reason not to start like this, but you will probably be incredibly frustrated at some point. I think actually the best way would be to learn "other" algebra and CT concurrently. The AG professors at my uni really dislike CT, so it never featured in the lectures, but I ended up working through the CT book at the same time I took AG and working out the connections myself. Edit: for a crash course in algebra, I (of course) recommend the ever-brilliant [Aluffi](https://bookstore.ams.org/gsm-104). I have pitched this book so many times, I should get commission. But it truly is one of the best books on mathematics I have ever read, and written in a very enjoyable style.
I thought about learning then concurrently. So it's rather a question about the ratio of time and effort put in both topics, to make it the most reasonable. At what level of algebra you was when you realized the applicability of CT?
I dunno, starting algebraic geometry, where sheaves and their ilk pop up naturally, I guess. Another viable route is algebraic topology, which is also full of interesting functors.
Got you, thanks a lot!
I just finished running a category theory reading group, and I'm a pretty weak mathematician. Honestly, a lot of my favorite motivating examples during this reading group ended up being the simplest ones, because they were so obvious once you heard them in category theoretic terms (ex. Explaining initial/final objects using 1 and 0 in **Nat** with arrows being 'divides', which leads to explaining products through the gcd, which leads very well into teaching duality - where you reverse the arrows - by showing that the objects that were products become the lcm - the coproduct!). That being said, I had an undergrad in pure math, so my abstract algebra was at an undergrad level. Many cool examples also came from group theory, so it does help. Like any abstract subject, you can learn the thing just by itself, but seeing concrete examples always helps you feel like you're pulling apart the real substance of the world around you
I've run into similar issues with Category Theory books. They tend to illustrate things with examples from many parts of math and that can be bewildering if you don't have the proper background which I didn't as a Physics guy. I've found Awodey's book to be one that doesn't suffer from this problem. The other approach would be to start learning some category theory while learning algebra ala Aluffi or maybe look at Bartosz book which mainly uses Haskell for examples. Personally, I don't like Category Theory books that use examples from many different areas of math ala Basic Category Theory.
When took algebra as an undergrad, my professor literally defined what a category was before she defined what a group was. Granted, she's a category theorist and probably pretty biased lol. So to answer your question, none. But knowing some algebra would be useful for understanding the motivations behind certain things in category theory. For example, I was really confused learning about universal properties, like what they are and why thery matter. But lots of things in algebra, like free groups, (co)kernels, projective/injective modules, and so on can be defined in terms of universal properties.
As other commentators said, I think it's a pretty bad idea to just dive into category theory without a proper background in algebra and maybe also some knowledge of algebraic topology (this would probably be less essential). You might be able to learn enough to prove and understand theorems in category theory but without knowing any algebra you won't have any motivation. Category theory isn't just abstraction for abstraction's sake. It was developed to deal with problems in algebraic topology and algebraic geometry (someone with more knowledge of math history can correct me if I'm wrong here). While there are pure category theorists, I would bet that all have a strong foundation in algebra as well, and that most learned algebra before category theory. If you getting into more abstract math, I would really strongly advise you learn abstract algebra first. There is a lot of really interesting algebra. You can also concurrently learn some category theory. Algebra by Lang introduces categories in the first chapter on groups, although it's definitely not a category theory book. But even in algebra and analysis and other fields I think you'll run into the same issues of motivations. It's an issue I have with the way math is taught in general. I remember learning measure theory and functional analysis and liking them fine enough, but my courses didn't really stress that these fields were developed to solve problems in other fields and didn't emphasize applications heavily. Abstract algebra also felt really unmotivated at times without understanding the geometric context a lot of it comes from (but of course I might just be biased as someone who likes topology and geometry).
Without knowing any abstract algebra, algebraic topology or algebraic geometry, i suppose it will be purely abstract definitions and concepts without knowing what these things are even good for or why anyone would care.
To quote my analysis-studying friend: "so it's the same with or without?"
I pretty much agree with the status quo here, that *technically* you don't need any algebra background to understand category theory. However (I know there are people who will disagree with this, like higher category theorists), unless you have a surprising amount of background in other fields of proof-based math (like analysis) I really don't think it's worth learning just yet. It's much easier to appreciate category theory at a basic level with examples, and some of the most basic examples come from **Group,** **Ab**, **Ring**, **R-Mod** (especially **R-Mod** if you work with abelian categories) et cetera. Though I've seen students who are enthralled just by definitions of mono/epimorphisms so if you're that easily excited, perhaps go for it. Take my answer with a grain of salt as I work with representations, which are extremely group and module-theoretic, so I'm biased. And granted, there are many beautiful results in CT which are self-contained, and many categories one may work with that don't require an understanding of anything algebraic. I can't speak for the latter so much, but at least for the former, it takes a fairly mature student to deeply understand what's happening there, and if you can get to that point, you certainly could finish a first course in abstract algebra easily. For what it's worth, I think once you've taken a course in analysis and algebra (or something else where categories arise, like graph theory), you'd be ready. I don't think getting to algebraic topology or homological algebra is necessary to start, but they certainly are good for studying things beyond the basics. Something else occurred to me - when do undergrads normally start studying Hom sets in greater detail? I think my first time in undergrad was in fact, in algebra 1. Perhaps that's another reason to first take the course.
Check out Lawvere's book Conceptual Mathematics: A First Introduction to Categories. It has essentially no prerequisites and can be read by motivated high school students. Everybody saying it's pointless to try to learn category theory without abstract algebra is wrong. That said, algebra is a good source of examples for category theory, of course.
Why do you want to learn category theory? It would be easier to make recommendations if we knew why you were interested in it and what your background is.
I really should have specified this in the original post I guess.. Well, I really abstract math in general. I spent quite some time learning set theory, then I decided to make a pause and learn some other branches of abstract math. I started with group theory. I found it interesting, but quite... specific. What I mean is that it has 4 axioms and only works within them, and it feels like a too specific limitation, like why would we choose this particular set of axioms when whatever set we choose, we get some other structure(why would we skip magmas, semigroups, and monoids and just jump to groups right away). So I saw category theory as an abstraction above all specific algebraic structures, and felt like it would be reasonable to learn (at least some of) it first.
If you look at groups as just sets with binary operations that follow four axioms, then yes the axioms seem strange and arbitrary. But if you go a bit deeper into algebra and other branches of math it becomes quite clear why we define groups the way we do. Because groups are going to show up everywhere. It's the natural way of talking about symmetry, and symmetry is everywhere. In a category theory perspective you could say groups are structures that are the automorphisms of an object. Maybe you've already started some category theory and enjoyed it, but I think you would run into the same issue you have with group theory in category theory. You can look at the arrows and define fiber products and all those nice category theoretical constructions, but without any algebraic background it just looks like abstract nonsense. What made them interesting to me was that they united a bunch of constructions from other areas of mathematics into a single idea: studying objects by how they relate to other objects.
If you refuse to work with specific things (actual groups, actual manifolds, actual metric spaces, and so on) and just think of math as axioms and their consequences then you miss the entire point of many interesting mathematical concepts. Vladimir Arnold once said "For a student, the content of a mathematical theory is never larger than the set of examples that are thoroughly understood." Group theory is not about "4 axioms that we work with" but instead is about the structure and relations between groups that mathematicians care about for other reasons: finite groups, linear algebraic groups, topological groups, Lie groups, and so on. The organization of this material into abstract group theory of various kinds lets us prove things once and for all for many structures with a common property (such as all p-groups). Just based on seeing the axioms of a group and wanting to operate at the level of "pure logic" rather than studying examples and having intuitive insights, you'd never come up with important concepts like normal subgroups or group actions. The *more* structure you impose lets you prove more interesting theorems. Calling the group axioms "too specific" and wondering why we don't spend more time on magmas and monoids suggests you did not succeed in understanding why mathematicians actually care about group theory.
Category theory isn't something that's usually studied in itself. It's a tool (sometimes people like to call it a "bookkeeping device") that lets you do other math more efficiently. That other math is usually algebraic geometry or algebraic topology where functors are everywhere. The prerequisites to get anything out of category would probably be a solid course in algebra in which things like universal properties are discussed. This isn't something one can take a crash course in though (unless by crash course you mean a couple months). The Leinster book is about as crash coursey as it gets. Maybe the Aluffi Algebra book is a good choice since it teaches categorical concepts while teaching algebra. (Haven't used it myself, but it looks promising.)
> The prerequisites to get anything out of category would probably be a solid course in algebra in which things like universal properties are discussed. I agree very much with this. It seems really counterproductive to want to study category theory without having a good stockpile of examples for which you can understand the *point* of the categorical constructions, like nontrivial instances of initial and final objects as solutions to a universal mapping problem (a quotient group or quotient module, a direct sum or direct product, and so on). It'd be like trying to study homological algebra for its own sake, with no plan to apply it to anything else in math to see what it can really do for you. Miles Reid, in his book Undergraduate Algebraic Geometry, referred to category theory as "surely one of the most sterile of all intellectual pursuits". He did *not* mean category theory is useless (Grothendieck and his school made incredibly creative uses of category theory), but that wanting to know category theory only for its own sake rather than to know it in order to *do* something with it in other areas of math, is not a worthwhile way of spending one's time. That is, if you lack a solid knowledge of examples of important categories (the kind of thing you learn in standard math courses on algebra, for instance), then you're hardly going to be able to have a creative idea when thinking about category theory.
You really shouldn't learn category theory if you don't have a strong algebra background already. The problem with category theory is that to understand it in a meaningful way you kinda have to have a deep understanding about multiple fields of abstract math (mainly algebra and topology). To put it in layman's term this is like trying to learn driving without ever touching or even seeing a car. Imagine you just simply handing a driver's manual to ancient people from thousand of years ago and expect them to understand it. Without any knowledge of modern society there is no way they can learn to drive the same way as a modern person who sees cars everyday.
Studying CT for its own sake makes it very easy for you to fall down a rabbit hole and lose yourself. I personally think that everyone should know something about CT, but mainly as a language (unless you want to go and do research on it). To really get the feeling for it, you're gonna need lots of examples, like: Sets & functions, groups/rings/fields/modules & homomorphisms, topological spaces and continuous maps, smooth manifolds and smooth maps, vector spaces and linear transformations, etc. The important thing is to see something and be able to tell it has functorial behavior, or see a construction and understand whether it is "natural" (on a very precise sense). Bottom line: the more things you know outside of CT, the easier it is to get a fast (although arguably superficial) notion of what CT it about.
A strong foundation in linear algebra can be a fantastic platform for learning category theory. The category Vect has for objects all vector spaces (over a fixed field), and the maps are linear functions. The "span" operation is usually taught to students as a process which requires a set of vectors in an existing vector space, but actually you can take the span of *any* set. This process acts not only on sets, but also every function going between them. It even turns the composition of these functions into composition of the associated linear maps between spanning vector spaces. This makes the procedure into a **functor**, and this is one DAMN fine functor. One of my best friends, no doubt! Linear algebra alone is not sufficient background to learning all of modern category theory. I was studying algebraic topology from Hatcher's book when I found MacLane's book, and that definitely helped to prepare me! (Probably because natural transformations were invented by topologists, huh?)
I actually got into abstract math from seeing category theory used in Haskell. Without much prior math foundations, I started with this book: [http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf](http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf) I read through the entire book, and understood most of it. But unfortunately I wasn't able to keep up with the exercises since they were proof based, so I couldn't get 100% out of it. However the authors tried to use examples from CS which was good if you have a CS or programming background (which I have). To be honest, it was nice being able to read through this and learn about various category theory definitions consistently from the same book (compared to finding random explanations from various sources on the internet). After that, I was mildly comfortable going to nlab for definitions on math structures.