A category is a collection of objects with a collection of morphisms with the following properties

1. Each morphism has a domain and codomain . We write .
2. Each object has an identity morphism .
3. For any morphisms there exists a composite morphism
4. For any , both and are equal to .
5. For any composable triple of morphisms , , , the composition is associative, e.g . We will denote it by

We will often call morphisms arrows.

We call a category small if it's morphisms fit in a set.

A category is locally small if for any two objects, the morphisms between them fit in a set.

A monoid is a one object category.

A preorder is a set with a binary relation , that is reflexive and transitive.

A preorder is precisely a (small) category in which there are no parallel pairs of distinct morphisms between any fixed pair of objects.

An isomorphism is a morphism for which there exists a morphism such that and . In that case we say that and are isomorophic , in symbols .

For a morphism and a left inverse such that is an identity, we call a retraction of f.

For a morphism and a right inverse such that is an identity, we call a section of f.

the left inverse and right inverse to a common morphism coincide. This means every morphism can have at most one inverse isomorphism.

Proof. TODO

A morphism with equal domain and codomain is called an endomorphism.

An endomorphism that is also an isomorphism is called an an automorphism.

A groupoid is a category in which every morphism is an isomorphism.

For a category , we call the collection of isomorphisms in the maximal groupoid inside .

A subcategory of a a category is definded by restricting to a subcollection of objects and subcollection of morphisms in , the identity morphism of any object in and the composite of any composable pair of morphisms in .

Let be a category. The maximal groupoid inside defines a subcategory.

Proof. TODO

For a category , the opposite category is given by the same objects and morhpisms as in , where every morphism in is defined to have opposite domain and codomain compared to the corresponding morphism in . For each object , serves as the identity in

We define composition to be . Observe that and are precisely composable when and are composable.

Note that for every category , its opposite category fullfills all axioms of a category.

Proof. TODO