Category

A category consists of

1. a collection of of Objects
2. a collection of Morphisms
3. two operations and sending morphisms to objects
   we write to mean and
4. An operation sending to where
5. A partial binary operation on morphisms such that:

and then and
subject to:
6. and whenever the composites are defined
7. Associativity: whenever and are defined

Isomorphism
Small Category
Quotient Category
Opposite Category
Functor
Skeletal
Balanced

Examples

Category of Sets
Category of Relations
Category of Partial Functions
Category of Groups
Category of Rings
Category of Vector Spaces
Category of Topological Spaces
Category of Metric Spaces
Category of Smooth Manifolds
Category of Topological Groups
Category of Homotopy
Category of Matrices
Monoid
Groupoid
Preorder Category
Cateogry of Small Categories