### Derived categories for dummies, Part I

#### Posted by Urs Schreiber

I am still trying to learn about derived categories, mostly using

P. Aspinwall: D-Branes on Calabi-Yau Manifolds hep-th/0403166 (2004).

For my own good here I’ll try to review a couple of key ideas. Corrections are welcome.

First of all: what is a **derived category** anyway?

I realize that there is lots of mathematical literature on this question which I am totally ignorant of. The following is what I extracted from Aspinwall’s discussion.

The short answer is: *The derived category $D(\mathcal{C})$ of any abelian category $\mathcal{C}$ is the category of complexes in $\mathcal{C}$.*

An **abelian category** is a category equipped with the necessary structure so as to make it possible to have sequences of morphisms in that category which form a *complex*. So in an abelian category $\mathcal{C}$ we can have sequences

with all the ${\mathcal{E}}^{n}$ being objects and all the ${d}_{n}$, $n\in \mathbb{N}$ being morphisms in $\mathcal{C}$ and such that the composition of any two subsequent morphisms ‘*is zero*’ in a sense.

Furthermore, we can speak of the n-th **cohomology** group

of such a complex. This is essentially the usual definition of cohomology suitably formulated so as to apply to general abelian categories.

These complexes ${\mathcal{E}}^{\u2022}$ are the objects of the derived category $D(\mathcal{C})$. There is an obvious notion of morphism between such complexes, and these will indeed be morphisms in $D(\mathcal{C})$. But $D(\mathcal{C})$ has *more* morphisms than just the obvious ones, and that takes a small amount of preparation to define:

There is an obvious and natural notion of a morphism between two complexes known as a **chain map**

which consists of morphisms

such that every diagram in sight commutes.

There are 2-morphisms between chain maps known as **chain homotopies**. Two chain maps related by a chain homotopy are ‘essentially the same’. All this is precisely as in the ‘ordinary’ (non-category theory) context. (Maybe if I find more time later I’ll spell out more details here.)

From any chain map $f:{\mathcal{E}}^{\u2022}\to {\mathcal{F}}^{\u2022}$ between the complexes ${\mathcal{E}}^{\u2022}$ and ${\mathcal{F}}^{\u2022}$ one gets an induced map

between the cohomologies of these complexes.

Note that ${f}_{*}$ contains less information than the $f$ it comes from. Due to that, ${f}_{*}$ may be invertible (be an isomorphism) while $f$ itself is not. In this case $f$ is called a **quasi isomorphism**. In the derived category $D(\mathcal{C})$ we *declare* these quasi-isomorphisms to be true isomorphisms, i.e. we declare that the morphisms in $D(\mathcal{C})$ are

1) all chain homotopy equivalence classes of chain homotopies between complexes

2) and all formal inverses of all all quasi-isomorphisms in 1).

That’s it. A derived category.

The crucial point of derived categories is that $D(\mathcal{C})$ knows how to **subtract** or **annihilate** objects of $\mathcal{C}$. $\mathcal{C}$, being abelian, is an additive category, but need not have a notion of subtraction.

In order to get that, one needs to make use of the **translation functor**

which shifts all the indices $n$ of the complexes by one $n\to n+1$.

Given two complexes, we can take their direct sum

But given a morphism

we can replace the map

with the map

Let me denote the result of this operation as

This corresponds to subtraction because, as one can show

is isomorphic to the trivial 0-complex (which is the 0-object in $D(\mathcal{C})$).

This has a very nice physical interpretation:

D-branes carry gauge fields which live in fiber bundles which can be described by their sheaves of sections. If $\mathcal{C}$ is the category of (‘coherent’) sheaves then the objects in $D(\mathcal{C})$ can be shown to describe general configurations of D-branes, up to some technical fine print. If ${\mathcal{E}}^{\u2022}$ describes some D-brane then $T({\mathcal{E}}^{\u2022})$ describes its anti-D-brane and

describes the mutual annihilation of the two, whith the off-diagonal map $f$ descibing the tachyonic strings which stretch from ${\mathcal{E}}^{\u2022}$ to $T({\mathcal{E}}^{\u2022})$ and which communicate the annihilation process.

So much for now, typing this stuff always takes longer than expected, even when already expected to take somewhat longer. More later, if time permits.

Here is a **question** that I would like to know the answer to:

Consider a quiver $Q$ and its category $\mathcal{C}=\mathrm{Rep}(Q)$ of representations. $\mathcal{C}$ is a 2-algebra. Is the derived category $D(\mathcal{C})=D(\mathrm{Rep}(Q))$ still a 2-algebra in an appropriately weakened sense? In particular, does the tensor product in $D(\mathcal{C})$ distribute (up to isomorphism) over the subtraction operation ${ominus}_{1}$?

I’d expect this to be true, but I have not yet figured out the details. I think this might be important because given a dimensional deconstruction of a $(\mathrm{2,0})$ theory by means of a ‘large’ quiver $Q$ a natural guess for the vector 2-bundles expected to play a role in the $(\mathrm{2,0})$ theory are modules of some 2-algebra naturally associated to $Q$, and the obvious guess is to use $D(\mathrm{Rep}(Q))$ – instead of $\mathrm{Rep}(Q)$ as I guessed previously.

## MathML

For some reason, at least on my system, whenever I had a

\mathcal{C}in the above formulas what appears on my screen is just a question mark. I didn’t have this problem before. Sorry about that. If anyone knows how to fix it please let me know.