Most of the modern texts use category theory for algebraic topology rather than set theory. What are the pros and cons of both the set theory and the category theory in this formulation.
Moreover, is it necessary to use Category theory or set theory suffices for all the concepts.
Thanks in advance!!!
This is a great question!
An overview of the advantages and disadvantages of category theory are given in
https://www.researchgate.net/publication/267676443_Structuralism_and_category_theory_in_the_contemporary_philosophy_of_mathematics
See, also:
Three Varieties of Mathematical Structuralismt - ResearchGate
https://www.researchgate.net/publication/240953896_Developments_in_Mac_Lane_Set_Theory
Article Structuralism and category theory in the contemporary philos...
Article Developments in Mac Lane Set Theory
I interpret your question even wider than this as `why bother with category theory at all?'
At the naive level, the most useful thing about category theory is the language it allows. For example, defining a pre-sheaf using categorical language is easy as a particular functor... you can look this up yourself. Any meaningful constructions in mathematics should be `functorial' - meaning very roughly well behaved with respect to morphisms. The kinds of invariants your construct in algebraic topology should be functorial.
Category theory allows for a compact discussions covering all the elements and morphisms of that category. This is the common use of category theory, as a simplifying compact language, but it is more than that...
At a slightly more advanced level it allows you to use rather general constructions and notions within a category. For example (when it makes sense) products, limits, colimits etc.
For me personally, one great use of category theory is via generalised objects and internal homs, which make sense due to the Yoneda lemma - which is probably the most powerful yet deceptively simple result known to man! Without details, the lemma allows one to replace an object with its corresponding Hom functor. This can really help in algebraic geometry. When you have an object as a functor, you can think think about more general functors that play the same role, hence generalised objects. These generalised objects are essential in supergeometry, for example.
If your root question is `can I get by without categories' then as a modern mathematician I would say no. A basic understanding of category theory is needed as it will be the language of all your fellow mathematicians. If your root question is `can one work without categories', then I would say yes, in the sense that most mathematical constructions do not rely on anything deep from category theory. That said, the constructions are usually better understood in light of a little category theory.
I am sorry I have not really pointed to much directly in algebraic topology, but still I hope I have said something to worth to you.
@Peter Breuer
I always say that a little category theory can get you along way, but a lot of category theory won't always get you much further!
@Romeo P.G
For sure this is true. There are many examples from algebraic geometry. I often work with kinds of locally ringed spaces - think of just generalising the sheaf of smooth functions on a manifold - and these are not `set theoretical objects' in the sense they are not simply a set of points (together with a topology or something like that).
But of course, as ringed spaces morphisms can be defined as so can their category. The point is that the objects are not sets, though in the cases I deal with, the Hom sets are genuine sets.
Russell paradox states that we cannot consider a set of all sets without confronting a contradiction, though we can still consider a class (rather than set) of sets (or things) together with morphisms connecting them called categories which prove to be vital for studying various mathematical structures ( topology is no exception) by putting those structures in an abstract universal frame in which it is concentrated on morphisms connecting mathematical structures and the relations which characterize them rather than concentrating on their defining sets.
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