CT: Categories

In Category Theory, A category is a group of objects and morphisms.

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• 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.
• CT: Morphism
  • In category theory, a morphism is a transformation from one object to another. This can be within a category or between categories.
• CT: Object
  • In Category Theory, an object is a part of a category. It exists to define each end of a morphism.
• Video Series: Category Theory
  • categories
• Identity Morphism
  • Each object in a category needs at least a morphism which starts and ends at itself.
• 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 composition of morphisms in a monoidal category corresponds to an associative binary operation
  • For a monoidal category $M$ each morphism in $M(m, m)$ corresponds to the operation defined for the associated monoidal set on one of the elements.
• 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
  • Categories are collections of objects and morphisms which are easily represented by nodes and edges respectively. While not every graph is a category, every category can be described as a graph. It's also true that any graph can be made into a category.
• 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.