Explain derived categories to someone without a strong background in homological algebra [closed]

Wikipedia has ample information about derived categories but it is too sophisticated for me to take in.

The derived category D(A) of an abelian category A is a category D(A) together with a functor Q: Kom(A) -> D(A), where Kom(A) is the category of cochain complexes on A.

For example, if A is the category of modules over a ring R, then Kom(A) is the category of cochain complexes whose terms are those modules over R. (A cochain complex is a sequence of objects with maps between them, where the composition of any two is zero.)

The derived category D(A) must have a universal property:

If there is some other category C, and a functor F such that for any quasi-isomorphism f in Kom(A), F(f) is an isomorphism in C - then F: Kom(A) -> C factors through Q: Kom(A) -> D(A).

There are many concepts here I would like to develop my intuition for.

1. What is an abelian category?
2. What is the difference between chain complexes and cochain complexes, and why must the definition of derived functors be in terms of cochains?
3. What is so significant about chain complexes in general? What useful properties do these sequences of maps have, given that $f_{i + 1} \circ f_{i}$ is “zero”? Why is that condition so interesting or necessary?
4. What does it mean to say the composition of the maps is “zero”? In category theory, there are zero objects, but I don’t know of “zero arrows”. I believe it means that the image of $f_i$ is the kernel of $f_{i + 1}$, but haven’t seen why this should be called “zero”.
5. What is the difference between a “universal” property, vs. just a property?

In summary, the derived category of an abelian category is a category where for a functor $Q$ from $Kom(A) \rightarrow D(A)$, for any other category $C$ whose isomorphisms are in correspondence with those in $Kom(A)$, $Q: Kom(A) \rightarrow D(A)$ can be represented as the composition of two functors, $F: Kom(A) \rightarrow C$, and then some functor from $C$ to $D(A)$.

My question is, why is that property significant? I read it generalizes the idea of localization of a ring, but I don’t understand that well enough yet.

posted 10 months ago

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Julius H.‭

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