How Much Category Theory Should A Topologist Learn
What are the prerequisites for studying category theory?
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well, as always, I initially took a look at what wikipedia says. the idea of talking well-nigh general mathematical objects and arrows between them sounds pretty impressive and quite exciting to me, but just similar any other math stuff, the idea looks quite elementary and the examples that wikipedia gives are quite understandable only when you get seriously engaged with the subject you find information technology challenging. so I wanna know what are the prerequisites for studying category theory. I'm already familiar with group theory and take solved a considerable portion of Herstein'south issues (except the ones that it describes them as hard or very hard) and I feel comfortable agreement grouping theory main topics at present. I don't know much about rings but I've studied Linear Algebra. I oasis't taken Topology in the university yet simply I'm taking a Real Assay grade this semester and it seems easy to me. Can I start reading about category theory or I have to wait to become more than experienced and informed in mathematics?
Category theory doesn't have whatever prerequisites. The theory can exist understood by a (proficient) freshman in mathematics. The point, nonetheless, is that yous need many examples to run into why category theory is useful or to see why certain concepts are defined the way they are.
Correct now, you've seen three categories:
- Sets and functions
- Groups and homomorphisms
- Vector spaces and Linear maps
Category theory generalizes this situation. But you will maybe not see why sure generalization are useful...
If yous want to learn category theory, then it is useful to larn the examples concurrently with the chiselled language. Here are some books to help yous:
- "Arrows structures and functors. The categorical imperative" by Arbib
This book presents many examples in category theory. And information technology too presents the categories in a fashion an undergraduate should understand.
- "Algebra: Chapter 0" by Aluffi
This book presents algebra from the view of category theory. It'southward worth to read the categorical definitions behind things like group theory. Certainly worth a read!!
More than advanced books: (not advisable for studying correct now, only practise requite them a look)
- "Abstract and concrete categories" by Adamek, Herrlich Strecker
The book is freely available at katmat.math.uni-bremen.de/acc/acc.pdf
The book is rather encyclopedic, and might non exist suited for self-written report. Do check it out occasionally, if just for the rich examples in the book!!
-"Category theory for the working mathematician" past MacLane
The bible of category theory. Many people larn categories from this very book. It'southward a fleck advanced though
-"Handbook of chiselled algebra" past Borceux
It contains almost everything you every desire to know about categories. It is a three book book and really, really nice.
Category theory doesn't have any prerequisites. The theory can be understood by a (good) freshman in mathematics. The indicate, however, is that you demand many examples to come across why category theory is useful or to see why certain concepts are divers the way they are.Right at present, y'all've seen three categories:
- Sets and functions
- Groups and homomorphisms
- Vector spaces and Linear mapsCategory theory generalizes this situation. But y'all will maybe non see why certain generalization are useful...
If you want to learn category theory, then it is useful to larn the examples meantime with the categorical language. Here are some books to assistance yous:
- "Arrows structures and functors. The categorical imperative" by Arbib
This volume presents many examples in category theory. And it also presents the categories in a way an undergraduate should empathise.- "Algebra: Affiliate 0" by Aluffi
This volume presents algebra from the view of category theory. It'due south worth to read the categorical definitions behind things like group theory. Certainly worth a read!!More advanced books: (non advisable for studying right now, but exercise requite them a await)
- "Abstract and concrete categories" by Adamek, Herrlich Strecker
The volume is freely available at katmat.math.uni-bremen.de/acc/acc.pdf
The book is rather encyclopedic, and might not be suited for self-study. Practice cheque information technology out occasionally, if simply for the rich examples in the book!!-"Category theory for the working mathematician" by MacLane
The bible of category theory. Many people learn categories from this very volume. Information technology's a bit advanced though-"Handbook of categorical algebra" by Borceux
Information technology contains almost everything you every want to know about categories. Information technology is a 3 volume book and really, really nice.
Thanks.
I knew virtually 'category theory for the working mathematician' past Saunders McLane but I read somewhere that Information technology was a graduate textbook. Will knowing category theory aid me to sympathize advanced algebraic structures easier? the thought looks pretty clear and impressive but the question is -as you said- why information technology's useful? what things volition I acquire if I study category theory?
Cheers.I knew about 'category theory for the working mathematician' by Saunders McLane merely I read somewhere that Information technology was a graduate textbook. Will knowing category theory help me to understand advanced algebraic structures easier? the idea looks pretty clear and impressive simply the question is -equally you said- why it'south useful? what things will I learn if I report category theory?
Yous mustn't actually meet category theory as a mathematical theory, it's more than a kind of language. It'southward a very handy and cool kind of language in which almost of mathematics can be told.
Many students accept trouble with the concept of "tensor product". They don't quite see what it is and how to handle information technology. But once yous've seen categories, then you tin can grasp the tensor product quite easily: it'southward the coproduct in the category of algebra's. And it's an adjoint to the hom-functor.
Category theory will allow you lot to see connections betwixt dissimilar branches of mathematics and it will brand these connections rigorous. For instance, why is the products of groups divers as information technology is?? Categories (concrete categories actually) will answer these questions very neatly.
Knowing categories won't help yous with grouping theory specifically or topology or etc. But it will help y'all come across some connections between the subjects.
On an undergraduate level, category theory can be eliminated (still, I really prefer non to do that), but on a graduate level categories are very necessary. Most of algebraic geometry (for case) has been washed with category theory. Homology must exist washed by categories. Algebraic topology must exist done with categories.
the basic moral of category theory is that the maps are more important than the objects, so every time you learn a new definition, like vector space, spend even more fourth dimension studying the maps between them, i.due east. linear transformations.
and when you memorize the definition of a topological infinite, spend much more than time making upwards examples of spaces and of continuous maps between them.
And when you learn the definition of a functor, like the fundamental group, or homology, practice computing the induced transformation of maps, from continuous maps to grouping homomorphisms, and compute every bit many as possible. E.f. endeavour to empathize when the induced homomorphism is an isomorphism and when it is zip.
on the other mitt, almost nobody needs to know the definition of a category.
the thing is that this semester I'yard taking linear algebra, abstruse algebra, calculus III, differential equations and number theory and I'm afraid that studying category theory would distract me from my schooling (it surely volition assistance my education though). if I know that knowing category theory will assist me learn and understand mathematics meliorate I'll definitely start studying it at present because I've had a very hard time agreement tensors since years ago to even now, although I was in high school when I studied them out of curiosity about general relativity only I remember that I never fully understood tensors no affair how difficult I tried and fifty-fifty today I can hardly empathise advanced topics about tensors if I tin at all. so, what do you advise? Is information technology a wise decision to study category theory now?
Is it a wise conclusion to study category theory now?
I'd say yes. Categories will give yous a unified understanding of mathematics. It volition assist yous (a flake) in abtract algebra, topology and linear algebra. And information technology will certainly aid yous to see connections between those topics.
And so over again, I'g a category theory-lover 
I'd say yeah. Categories will give you lot a unified agreement of mathematics. It will aid you (a scrap) in abtract algebra, topology and linear algebra. And it will certainly help you to meet connections between those topics.And so again, I'chiliad a category theory-lover
I've found this book: Category Theory, by Steve Awodey past Oxford science publications. Information technology covers these topics. It covers these topics:
Affiliate 1. Categories
1.i Introduction
one.2 Functions of sets
1.3 Definition of a category
1.4 Examples of categories
one.5 Isomorphisms
1.six Constructions on categories
ane.7 Free categories
1.8 Foundations: large, small, and locally small
2. Abstract structures
2.i Epis and monos
2.2 Initial and final objects
2.3 Generalized elements
ii.4 Sections and retractions
ii.five Products
2.6 Examples of Products
2.vii Categories with products
2.8 Hom-sets
3. Duality
3.1 Duality principle
iii.two Coproducts
three.3 Equalizers
3.4 Coequalizers
4 Groups and categories
4.1 Groups in a category
four.2 The category of groops
4.3 Groups as categories
4.four Finitely presented categories
v. Limits and colimits
5.1 Subobjects
5.2 Pullbacks
5.3 Properties of Pullbacks
five.four Limits
5.v Preservation of limits
5.6 Colimits
6. Exponentials
6.1 Exponential in a category
6.2 Cartesian airtight categories
half-dozen.3 Heyting algebras
6.4 Equational definition
half dozen.5 Lambda calculus
vii. Functors and naturality
7.1 Category of categories
vii.two Representable structures
7.3 Stone duality
7.4 Naturality
vii.v Examples of natural transformations
7.vi Exponentials of categories
7.vii Equivalence of categories
7.8 Examples of equivalence
viii. Categories of diagrams
8.1 Gear up valued functor categories
8.2 The Yoneda embedding
viii.3 The Yoneda Lemma
8.4 Applications of the Yoneda Lemma
8.five Limits in categories of diagrams
8.6 Colimits in categories of diagrams
eight.vii Exponentials in categories of diagrams
eight.8 Topoi
nine. Adjoints
nine.1 Preliminary definition
nine.2 Hom-set definition
ix.iii Examples of adjoints
9.iv Order adjoints
9.5 Quantifiers equally adjoints
ix.six RAPL
9.7 Locally cartesian closed categories
9.viii Adjoint functor theorem
10. Monads and algebras
10.1 The triangle identities
10.2 Monads and adjoints
ten.3 Algebras for a monad
10.iv Comonads and coalebgras
10.five Algebras for endofunctors
Is information technology suitable for self-studying?
[reid, undergraduate algebraic geometry, p. 116.]
on the other mitt miles reid and everyone else knows what a product is, and a sum, and what duality ways, and what Hom and tensor are, and what projectives and injectives are, and initial and final objects, and kernels and cokernels, and quotients, and fibered products and changed and straight limits, and sheaves, and what is an adjoint functor, and what derived functors are.
its not entirely useless, only it is sort of like studying grammar instead of literature.
I would advise the little book by peter freyd, abelian categories, as a overnice brusque introduction.
for starters you might read section I.10 of my free algebra course notes at:
http://www.math.uga.edu/%7Eroy/843-i.pdf [Broken]
called "categories and functors: what are they?"
I used the concept in that course to motivate the definition of normal field extensions. I.e. once nosotros divers the notion of galois group of a field extension, categorical thinking made us ask what are the induced maps between galois groups induced by maps of fields. it turns out there aren't whatsoever unless the map of fields induces a normal extension., so we were naturally led to that concept as well.
then it is helpful to have the nearly rudimentary ideas of category theory to guide many other studies, but obsessing over detailed concepts such equally seem to be belabored in that book you lot reference, is for near of us a recipe for extreme boredom. other opinions do exist nonetheless.
e.g. yoneda'south lemma motivates a philosophy for how to define things in mathematics. it says that if you know all the maps in or out of an object, then you know the object.
I.east. if Hom(.,A) and Hom(.,B) define naturally equivalent functors, then A ≈ B. This is proved by
noting that the assumed isomorphism between Hom(A,A) ≈ Hom(A,B) sends the identity map 1A, to an isomorphism from A to B. Big deal. This was a homework exercise in my algebra class. [Hint: how practice y'all suppose y'all would go about finding its changed? If yous think of looking at the isomorphism Hom(B,A)≈ Hom(B,B), you get the thought.]
But the implications are that in a subject similar algebraic geometry, one way to define the moduli infinite, or universal parameter space<
of all algebraic curves is to tell what morphisms into information technology should exist. These should be parametrized families of algebraic curves, i.e. maps X-->B whose fibers are curves, i.e. a family of curves parametrized by a infinite B, should define a morphism B-->M from the parameter space B into the universal parameter space M.
whether or non such an M exists, and what are its properties, is another matter, and more than interesting.
Is it suitable for cocky-studying?
It seems to be freely available at teaguesterling.com/category-theory.pdf
It seems similar a very good book which introduces the nearly important topics in category theory and from a modern betoken-of-view. Starting with this volume will certainly be ok.
You might miss some basic examples. Nigh examples come from abstract algebra/algebraic geometry/topology. Since you only seen groups, it means that you tin merely refer to groups for examples and insights.
Concepts like limits, Yoneda embeddings, etc. might not be easy to understand in the beginning without much examples. It is worth knowing them considering they are a existent handy tool sometimes.
I would occasionaly check the "joy of cats" at katmat.math.uni-bremen.de/acc for interesting examples. I strongly disagree that category theory is boring or useless. Simply such opinions do exist. I call up nigh people have blackness-and-white visions on categories: they love information technology or they detest it. I recall it's obvious what kind of person I am
mathwonk; but obsessing over detailed concepts such as seem to exist belabored in that book you reference said:I was in a true cat reading grouping, and I'll 2d this. It gets dry very rapidly in one case the novelty wears off. You demand lots of examples and applications to keep going.
And that's actually the trick, I recollect, if you want to larn category without a lot of previous groundwork. Yes, the subject field itself will exist very tedious, if yous don't accept a lot of proficient examples and things to brand it fun. The way that usually happens is when you report enough algebraic topology or maybe some other subjects. But it is possible to come up up with more elementary examples. That merely isn't the usual way of motivating category theory.
I retrieve it may depend a lot on what you end up doing. Categories actually play a key role in my ain work. Mayhap one of the chief points of my PhD thesis is that you can brand things in the subject a lot more than elegant and understandable if you put them in terms of category theory. Many papers in the subject area are just a mess compared to my (well, not just mine) not bad way of phrasing things, which is all made possible by using categories. This might sound like a agglomeration of hot air, just if yous were to actually read the papers in question, the departure is just night and day, I recall. I wasn't particularly a fan of category theory when I first encountered information technology. But, I have found it has proved its worth. Mind you lot, this is coming from someone who is an extreme conceptual/intuitive/visual thinker--manner out there on the fringes of it. Not what you might look for such an abstract subject.
As well, at some point, yous'll want to look at John Baez's website (I especially recommend the seminar notes). Very impressive category theory ideas there, and much more than.
but obsessing over detailed concepts... is for most of the states a recipe for extreme boredom.
Really? I can't recall of anything more than worthy of obsession. It sounds admittedly lovely. Hmm, category theory...
Really? I can't retrieve of anything more worthy of obsession. It sounds absolutely lovely. Hmm, category theory...
While it looks like you are going to spend a lot of effort on category theory, near mathematics I have seen only uses what it needs of it. I have never seen category theory to be essential in itself - not to say that that can't happen.
This book groups and shows the relationships amid diverse mathematical areas. It also has a chapter on "Category Theory" in the context of these areas.
Past the way I am a Mechanical Engineer and Software Engineer, and afterward some inquiry that the best way to tie these concrete and abstruse fields of engineering is through is Category
it helps if you lot are a masochist. i wouldn't bother much with category theory, or mayhap i should say i wouldn't bother with much category theory. it is more useful to learn some mathematics.the basic moral of category theory is that the maps are more important than the objects, and then every fourth dimension y'all learn a new definition, like vector space, spend even more time studying the maps between them, i.e. linear transformations.
and when you lot memorize the definition of a topological space, spend much more time making up examples of spaces and of continuous maps between them.
And when you lot learn the definition of a functor, like the fundamental grouping, or homology, practice computing the induced transformation of maps, from continuous maps to group homomorphisms, and compute equally many as possible. E.f. try to understand when the induced homomorphism is an isomorphism and when it is zero.
on the other hand, nigh nobody needs to know the definition of a category.
If you are a mathematician then you lot are by definition a masochist.
Sorry for the necrophilia.
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