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