Identity Morphism

Each object in a category needs at least a morphism which starts and ends at itself.

Backlinks

• Category Theory
  • At the core of category theory is, unsurprisingly, the category. A category is a collection of objects and morphisms, where each object has at least an 'identity morphism'. A morphism is an arrow pointing from one object to another. Objects exist as named points to give the morphisms context. Category theory concerns itself with the composition of morphisms within different categories and the different states that are possible.
• A category with a single object is a monoid
  • Any category with a single object is a monoid. The object in a monoidal category can have any number of morphisms greater than 1 - the requirement for an identity morphism isn't relaxed.
• The identity morphism in a monoidal category corresponds to the unit element
  • The unit element defined for a set-theory monoid corresponds to the identity morphism within a monoidal category.
• Graphs can represent a category
  • A graph must be cyclic to describe a category due to the requirement for identity morphisms.
• Any graph can be made into a category
  • Graphs can represent a category. It's possible to turn a graph that doesn't into a category by adding more edges. First, ensuring that each node has an identity morphism, then adding an edge for each pair of adjacent edges to satisfy the requirement for composition.