category

Idea

• a collection of things and binary relationships between them, such that these relationships can be combined and include the “identity” relationship “is the same as.”
• a quiver with a rule saying how to compose two edges that fit together to get a new edge.
  • furthermore, each vertex has has edge going to itself, acting as an identify for this composition.
• a combinatorial model for a directed space: ...

Definitions

• : exists two broad ways to define
  • one definition generalizes well to the notion of internal category, while the other generalizes well to the notion of enriched category.
  • good to know both
  • they are equivalent
• 1. definition with one collection of morphisms
  • a collection \(C_0\) of objects
  • a collection \(C_1\) of morphisms
  • source / domain and target / codomain: for every morphism \(f\), an object s(f) and an object t(f)
  • composite: for every pair of morphisms \(f\) and \(g\), where t(f) = s(g), a morphism \(f \circ g\)
  • identity morphism: for every object \(x\), a morphism \(id_x\)
  • such that the following properties:
    • source and target are respected by composition
    • source and target are respected by identity
    • composition is associative
    • composition satisfies the left and right unit laws
      • if s(f) = x and t(f) = y, then \(1_y \circ f = f = f \circ 1_x\)

  ---

  • semicategory: if the identity-assigning map and its axiom is omitted
• 2. definition with a family of collections of morphisms
  • a collection \(C_0\) of objects
  • for each pair \(x\), \(y\) of objects, a collection \(C_1(x, y)\) of morphisms from \(x\) to \(y\)
  • composite: for each pair of morphisms \(f\) in \(C_1(x, y)\) and \(g\) in \(C_1(y, z)\), a morphism \(f \circ g\) in \(C_1(x, z)\)
  • identity morphisms: for every object \(x\), a morphism \(id_x\) in \(C_1(x, x)\)
  • such that the following properties hold:
    • composition is associative
    • composition satisfies the left and right unit laws

Alternative Definition

• single-sorted definition: a variant of the first definition
• type-theoretic definition: a variant of the second definition
• protocategory: a mixture of the first and second definitions

Equivalent Definitions

• a monad in the 2-category of spans of sets
• a monoid in the monoidal category of endospans on the set of objects
• a simplicial set which satisfies the Segal conditions
• a simplicial set which satisfies the weak Kan complex conditions strictly

Generalizations:

• Internal categories: define a category internal to some other category
• Enriched categories: define a category enriched over some other category
• Indexed categories: captures the idea of working "over a base" other than Set
• Multicategories: allows morphisms to go from several objects to a single object
  • aka operad

Examples: ...