Trying To Learn Category Theory

In summary, the conversation discusses the topic of learning Category Theory and a problem from the text "Basic Homological Algebra" by Osborne. The individual asking for help shows their solution and asks for feedback on where they may be going wrong. Two other individuals provide helpful comments, with one suggesting a more elegant solution using the Lemma (Yoneda) and the other offering a cleaner version of the original solution. The original poster expresses gratitude and suggests following the more elegant solution. The solution is attributed to Maurice Auslander and possibly Eilenberg and Macl
  • #1
nateHI
146
4
I'm trying to learn Category Theory; this isn't homework or anything. I've attached a problem from the text "Basic Homological Algebra" by Osborne and I show my attempt at a solution. My solution doesn't seem exactly correct and I state why in the attachment as well. Can someone take a look and let me know where I'm going wrong?

Thanks!
 

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  • #2
nateHI said:
I'm trying to learn Category Theory; this isn't homework or anything. I've attached a problem from the text "Basic Homological Algebra" by Osborne and I show my attempt at a solution. My solution doesn't seem exactly correct and I state why in the attachment as well. Can someone take a look and let me know where I'm going wrong?

Thanks!
I think your problems start with the definition of the ##S_i## where you hide ℤ somehow. What you really want to show is that for each ##x ∈ S## there is a copy of ℤ in ##A##. To "redefine" ℤ via ##|S_i|## seems artificial to me. What should be the elements of ##S_i##? Consider the free groups generated by a single ##x∈S## instead. The only thing that makes it all a little abstract is that you don't know how big ##S## is. You could try to prove it for finite ##S## first.
 
  • #3
To me category theory is the art of phrasing everything in terms of mapping properties and avoiding use of individual elements. So I would do this problem this way:

Lemma (Yoneda): An object in a category is determined by the Hom functor it defines, i.e. two objects A,B are isomorphic in any category if and only if the functors Hom(A, ), and Hom(B, ) are equivalent.

So if Frab(S) is the free abelian group on the set S, and if Copr(Z;S) is the coproduct of the family of copies of the integers Z indexed by the set S, we want to show the Hom functors Hom(F(S), ) and Hom(Copr(Z;S), ) are equivalent.

By definition of coproduct, we have Hom(Copr(Z;S), ) ≈ Prod(Hom(Z, );S) ≈
(*) Prod( ;S), where Prod is the product functor and Hom is homomorphisms in ab.

But by definition of the free abelian group, also Hom(F(S), ) ≈ Map(S, ) ≈ Prod( ;S), where Map denotes set functions, i.e. morphisms in the category of sets.

Since we have shown Hom(F(S), ) and Hom(Copr(Z;S), ) are equivalent, thus F(S) and Copr(Z;S), are isomorphic.

(*) Now in this step, i.e. the equivalence of Hom(Z, ) with the identity functor, we do need to use elements to prove e.g. for every abelian group G, that Hom(Z,G) ≈ G, because Z has been given to us as a concrete group, not in terms of its mapping properties.
 
  • #4
to answer your question about your solution, isn't your map phi:S-->sigma(A) the map they speak of from S to "A".
 
  • #5
Thanks mathwonk and fresh_42 for the great comments. I've read them and will rework this problem shortly. Seems like it needs a lot of work though so it might take a day or two.
 
  • #6
So I cleaned my solution up a little based on the comments but I also worked through mathwonks solution which is much nicer. It used some mathematics that were a little beyond where I'm at currently but I managed to get a handle on those mechanics by "reading ahead."

Anyway, if anyone is reading this I propose you follow mathwonks solution and ignore my clunky attempt as his seems to be the more elegant (and correct) method.
 
  • #7
thank you for your kind comments, it is so much more satisfying to answer a question when the OP reads the answer and responds as graciously as you did about the benefit he/she derived. this beautiful approach to the topic was explained to us by the great maurice auslander in his first year graduate algebra class at brandeis in 1965.
 

1. What is category theory?

Category theory is a branch of mathematics that studies the structure of mathematical objects and the relationships between them. It provides a formal framework for understanding and analyzing various mathematical structures.

2. Why is category theory important?

Category theory has many applications in mathematics, computer science, and physics. It provides a powerful tool for understanding and organizing complex mathematical ideas and structures. It also allows for generalization and abstraction, making it useful in a wide range of fields.

3. How can category theory be applied in computer science?

Category theory has been applied in computer science to study functional programming languages, design and analyze algorithms, and develop type systems. It also provides a useful framework for understanding databases and other computational structures.

4. Is category theory difficult to learn?

Category theory can be challenging to learn due to its abstract nature and use of advanced mathematical concepts. However, with patience and practice, it can be understood and applied effectively.

5. What are some good resources for learning category theory?

There are many textbooks, online courses, and video lectures available for learning category theory. Some popular resources include "Category Theory for the Sciences" by David Spivak, "Category Theory in Context" by Emily Riehl, and the "Category Theory" course on Coursera by Bartosz Milewski.

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