What are the important properties that categories are really abstracting?

I have read some things in category theory and watched some lectures. There is a part of some lectures given by Steven Roman, on YouTube. At these lectures, he talks a bit about the process of abstraction, from what he said and from what I have observed until now, lets take an example of abstraction: Groups.

It seems that the whole usage of the concepts of groups depends on two phases: Identification and inheritance of properties. We first name some axioms (in this case, the axioms of group theory) and without mentioning any specific group, we deduce things with the previously given axioms and with this, we can deduce an arbitrarily long chain of theorems. Now if we can identify that some mathematical object behaves like a group, then this mathematical object inherits all the chain of deduced theorems and this is good because the step of identification is really small $\tiny( \text{we just need to check the axioms})$ compared to what the step of identification would be if we had to check every axiom and theorem. So, here I have a small hint of why the abstraction is made because I know some theorems of group theory and I have a fair guess of why these theorems would be important.

Now, changing the example, we can jump to categories and this is where I get lost. I understand that we can deduce things in categories and for certain categories, everything that was deduced is valid for any suitable category, that is: If we identify something as a category, then it inherits all those theorems deduced from categories. But now, I have no clue of why abstraction is made, nor which theorems would be important nor how they go from category to Its materialization in concrete mathematical objects.

There is also another thing: The idea of comparing categories, this seems even more misterious. What do we gain from being able to compare categories? Are there examples of relevant/important categorical comparisons at the level of elementary mathematics? Say: Real/Complex Analysis, Topology, Algebra and Combinatorics at an undergraduate level? I'm sorry if this question is bothersome but I guess Its the most organized and honest bunch of questions I could make.

• I am not sure I understand what is meant by "materialization", "concrete mathematical objects", it might help if you could explain or link the definition. – user337830 Nov 16 '16 at 16:10
• @user337830 More or less as Awodey says in his book, Page 2: "The important notion of adjoint functor occurs in logic as the existential quantifier and in topology as the image operation along a continuous function". What I meant with materialization is the occurrence of the adjoint function as those concepts. And these concepts are the concrete mathematical objects. – Billy Rubina Nov 16 '16 at 19:54

Consider the following objects, which I allow myself to make brief and imprecise descriptions in order not to make this answer painfully extense:

1. The Stone-Čech compactification - Given a topological space, we want to make a big compact space which extends sufficiently many functions.
2. The quotient of a vector space - We identify vectors on a given subspace.
3. The quotient of a topological space - We identify points on a topological space, "gluing" things together.
4. The completion of a metric space - We want a complete space which has as a dense subspace our original metric space.
5. The tensor product - We transfer multilinear information to linear information.
6. The universal cover - See here.
7. The free abelian group generated by a set - We want to generate a "formal" abelian group which has as generating elements the elements of a set.
8. etc (free group, direct limits, inverse limits,...)

All those examples, which may seem disjoint at first, are all an instance of the concept of "universal property", having defined the adequate categories.

For examples of functors, we have:

1. Homology.
2. The matricial representation of linear maps.
3. The fundamental group.
4. The chain rule.
5. Dualization of vector spaces (and more generally of modules (and more generally $Hom$ of a category itself))
6. Tensorization
7. Presheaves
8. etc

What point am I trying to make? The point is: the important property that categories are really abstracting is structure itself. Category theory endows a mathematician with a capability to soar through the fields of mathematics with a broad vision. More than "inheritance of properties", there is identification of a common underlying structure. The naturality of such idenfication comes with time and experience.

And all the above is an understatement. The distillation process which category theory provides is a powerful tool in several aspects. For example, it is common to have a situation where we want things to exist and they are imposed by some condition. And the imposition commands the definitions, and proofs are matter of routine checking. This is true, for example, in the isomorphism theorem. And the list goes on.

• Thanks for your reply. What I am curious now is that it seems that when we find a functor from one category to another, then there are things that can be done in the first category to prove something in the second. I'm trying to learn about why this is so. Are you aware of some very basic example on this? I'm sorry if the answer to this can be given in your answer, but I just don't see at the moment. – Billy Rubina Nov 19 '16 at 14:54
• @OppaHilbertStyle A basic example could be for instance the fact that $D^n$ does not retract to $S^n$. Because assuming it could, you would have maps $i: S^n \to D^n$ and $r: D^n \to S^n$ such that $r \circ i=Id$. Applying the homology functor, you would have the identity between $\mathbb{Z}$ and $\mathbb{Z}$ factoring through the trivial group, which is an absurd. Therefore, you have a statement in the topological category proved by a statement in the abelian groups category by means of the homology functor. This is the essence of algebraic topology, for example. – Aloizio Macedo Nov 19 '16 at 15:06