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

**Closed**
as too generic
by Peter Taylor
on Feb 19, 2024 at 08:25

This post contains multiple questions or has many possible indistinguishable correct answers or requires extraordinary long answers.

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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.

- What is an abelian category?
- What is the difference between chain complexes and cochain complexes, and why must the definition of derived functors be in terms of cochains?
- 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?
- 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”.
- 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.

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