An additive category is a category such that:

which is a coproduct:

and also a product:

The category of projective -modules is an additive category.

Let be an additive category. Let be a morphism in .

If a cokernel exists, it is the pair satisfying the universal property:

Therefore,

If a kernel exists, it is the pair satisfying the universal property:

Therefore,

Suppose every morphism has a kernel and cokernel.

There is a canonical map given by described belowbelow.

There is an isomorphism by the 1st isomorphism theorem.

We have

In particular, there is a canonical element : it satisfies the listed properties by putting and in the definitions of kernel and cokernel above. (Done in Assignment 3.)

The canonical element is what induces the isomorphism in the st isomorphism theorem. It is not an isomorphism in general.

Let . We have , but there is a map

We can check that and , and we find that , . The map

is not an isomorphism in (the inverse is not continuous).

An abelian category is an additive category such that every morphism has a kernel and cokernel, and for all , the canonical map is an isomorphism.

Let be a topological space. The category of sheaves on is “obviously” additive.

Recall that a sheaf is:

and if is an open cover:

Let be a morphism of sheaves. Then

is a sheaf. Exercise: check this.

On each open set , we have

where the existence and uniqueness of the map comes from the universal property of kernels in the category of abelian groups (which are indeed the kernels in the category theoretic definition).

The definition

is not always a sheaf. We have to sheafify:

where denotes sheafification. Recall that we had the adjoint pair

Therefore,

giving us

where the map uniquely determines the map by adjunction.

The picture:

In this diagram and require sheafification. So when we are describing the kernel of the morphism , we will also need to take sheafification into account.

We want to know if . For , working with an explicit model of sheafification, we have

On the other hand, consists of , where is an open cover, plus gluing. One sees that the condition is the same as the condition by the 1st isomorphism theorem for abelian groups.

The category of sheaves on a topological space is an abelian category.

Prove that are not many projective objects in .

Let be a continuous function.

We define

This definition makes sense because if is an open then is open.

This is harder to define: we can’t just put in because it may not be open. Instead, we define

is left adjoint to , i.e.,

An element is a family of maps

plus compatibility with restriction.

An element is a family of maps

But to map out of a direct limit, we need to specify a map for every (as depicted by above).