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 $1_y \circ f = f = f \circ 1_x$

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  • 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: …