2024-07-23

Definition (Category)

A category consists of the following data:

  1. A class of objects .
  2. For each , a class of morphisms from to .
  3. For every , a binary operation

The above data should satisfy the following:

  1. (Associativity) for all , , .
  2. (Identity) For all , there exists such that if then .

Definition (Locally small)

A category is called locally small if is a set for all .

Examples of categories

Objects Morphisms Composition
Sets Functions Composition
Topological spaces Continuous functions Composition
Abelian groups Group homomorphisms Composition
( field) -vector spaces Linear transformations Composition
For a topological space, Open sets in Inclusions: if , if
Any poset Elements Ordering Transitivity
A monoid 1 object Locally small
A group 1 object Locally small and invertible
( unital algebra) -modules Linear maps Composition

Definition (-modules)

Here, we think of an algebra as a set with operations such that is an abelian group and satisfies

  1. .
  2. and
  3. There exists such that for all .
    No commutativity assumed.
Example 2024.1 (Matrix algebra).

For example, is an algebra for any field .

An -module (left -module) is an abelian group with a binary operation , such that

  1. for all .
  2. for all ,
  3. and for all , .

Example

In the case , a -module is the same thing as an abelian group. In the case a field, a -module is the same thing as a -vector space.

Definition (Morphisms of -modules)

#TODO2
A morphism of -modules is a map

between -modules and such that

for all , .

More examples of categories

Objects Morphisms Composition
( unital algebra) -modules Linear maps Composition
for a category Objects of for each : Reversed composition

A final example

Let be a field. Let the objects be for vector spaces , over , and a linear transformations. A morphism between two objects and is a pair of morphisms and such that :

Exercise 2024.2 (Quiver algebra).

This example can be realised as an -module for a well-chosen algebra .

Solution. Define the quiver algebra of the quiver

over the field to be the algebra

Then the category is equivalent to the above example by associating to each object the -vector space with the -module structure given by

A morphism is then sent to the -linear map . □

Definition (Functor)

Let and be two categories. A functor from to is the following data:

  1. For all , assigns to .
  2. For all , assigns to .

The above data should satisfy the following:

  1. for all .
  2. for all , which are composable.

Examples of functors

  1. , , and . Here, , and .
  2. Let be a locally small category and . Let , and define . Given , we get
    #TODO diagram
  3. What about ? This is a functor from to .
  4. Let be the category defined above. Define by
    where . Exercise: Define on morphisms and check that it is a functor.