2024-10-01

Towards derived categories

Derived categories: a place for homological algebra.

Something simpler

Let be an additive category. For example, projective objects in an abelian category. Let be the category of complexes where objects and differentials are in : an example of an object looks like

Some subcategories are:

  • , the bounded below complexes, i.e., for .
  • , the bounded above complexes, i.e., for .
  • , the bounded complexes.

Homotopy category

Recall that are homotopic if there exists such that

¢¢¢A¡1A0A1¢¢¢¢¢¢B¡1B0B1¢¢¢ddfgdhfgdhfgdddd

Philosophy: Homotopic maps should be the same. #TODO2

We define the homotopy category :

  • The objects are the same as in .
  • The morphisms are
    Need to check that this is an group: we are quotienting by a subgroup (abelian) because given two maps that are homotopic to , we can just add their s.
  • Composition: Pick chain map representatives and compose, then go back down. We need to check that this composition is well-defined.

The check for composition being well-defined boils down to checking the following: given

we must check

  • If then .
  • If then . Let’s check this one. We have
    Since is a chain map, and commute #TODO diagram, so

We can also consider , , just as for . We conclude that , where , is a category.

  • is additive.
  • is not abelian (maybe later).
  • is triangulated.

Definition (Mapping cone)

Let be a morphism of complexes. Its mapping cone is the complex , where

The differential is #TODO diagram

Claim: This is a complex.
Proof. We just square :

because is a chain map.

Note that the complex is shifted to the left:

Remark

The above construction is done in the category of chain complexes , rather than in the homotopy category . However, if we take a homotopic , we get a priori a different object in , but it turns out to be an isomorphic object.

Definition (Shift)

Let be a complex. Define the shifted complex , , to be

Note that a positive integer shifts the objects to the left: for example

Remark: Mapping cones again

There is a short exact sequence of chain complexes

Notice that the negative sign on in the definition of the differential for matches with the negative sign in the definition of the differential for , for otherwise we would not have a short exact sequence of chain complexes.

Q: What happens if we take the mapping cone of a composition of complexes?
A: The octahedral axiom.

Definition (Triangle)

A triangle is an exact sequence

We say that the triangle is distinguished if it is isomorphic (in ) to

Remark

Because shifting is a functor, we automatically get a long exact sequence

XYZX[1]Y[1]Z[1]X[2]Y[2]Z[2]¢¢¢

Axioms

The collection of distinguished triangles “replaces” the notion of an exact sequence. They satisfy

  1. (TR0) A triangle isomorphic to a distinguished triangle is distinguished.
XYZX[1]X0Y0Z0X0[1]»=»=»=»=
  1. (TR1) For all ,
    is distinguished.
  2. (TR2) For all , there exists a distinguished triangle
  3. (TR3) Triangles can be rotated. If
    is distinguished, then
    is distinguished.
  4. (TR4) If the horizontal sequences are distinguished triangles and we have commutative squares:
XYZX[1]X0Y0Z0X0[1]fg9'f[1]

then there exists such that all squares commute.
5. (Octahedral axiom) Idea: We want to compare , , and :

Y0Z0X0XZYÃ+1'+1g±ffg+1+1

We are given , yielding the bottom commutative triangle. This leads to the three solid distinguished triangles by (TR2) (we think of , , ). The octahedral axiom states that there exists a distinguished triangle on top, where the maps in the distinguished triangle arise in the following way:

  1. For the distinguished triangles, (TR4) gives the existence of :
XYZ0X[1]XZY0X[1]'
  1. For the distinguished triangles, (TR4) gives the existence of :
XZY0X[1]YZX0Y[1]fÃf[1]

These maps must fit in the commutative octahedron above.

Definition (Triangulated category)

An additive category is triangulated if there exists an autoequivalence , , plus the datum of a collection of distinguished triangles

satisfying the axioms above.

Example

. #TODO2